1300 Math Formulas

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1300 Math Formulas - Alex Svirin

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Page 1: 1300 Math Formulas
Page 2: 1300 Math Formulas

1300 Math Formulas ==============================

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Preface ====qÜáë= Ü~åÇÄççâ= áë= ~= ÅçãéäÉíÉ= ÇÉëâíçé= êÉÑÉêÉåÅÉ= Ñçê= ëíì-ÇÉåíë= ~åÇ= ÉåÖáåÉÉêëK= fí= Ü~ë= ÉîÉêóíÜáåÖ= Ñêçã= ÜáÖÜ= ëÅÜççä=ã~íÜ=íç=ã~íÜ=Ñçê=~Çî~åÅÉÇ=ìåÇÉêÖê~Çì~íÉë=áå=ÉåÖáåÉÉêáåÖI=ÉÅçåçãáÅëI=éÜóëáÅ~ä=ëÅáÉåÅÉëI=~åÇ=ã~íÜÉã~íáÅëK=qÜÉ=ÉÄççâ=Åçåí~áåë= ÜìåÇêÉÇë= çÑ= Ñçêãìä~ëI= í~ÄäÉëI= ~åÇ= ÑáÖìêÉë= Ñêçã=kìãÄÉê= pÉíëI= ^äÖÉÄê~I= dÉçãÉíêóI= qêáÖçåçãÉíêóI= j~íêáÅÉë=~åÇ= aÉíÉêãáå~åíëI= sÉÅíçêëI= ^å~äóíáÅ= dÉçãÉíêóI= `~äÅìäìëI=aáÑÑÉêÉåíá~ä=bèì~íáçåëI=pÉêáÉëI=~åÇ=mêçÄ~Äáäáíó=qÜÉçêóK==qÜÉ= ëíêìÅíìêÉÇ= í~ÄäÉ= çÑ= ÅçåíÉåíëI= äáåâëI= ~åÇ= ä~óçìí= ã~âÉ=ÑáåÇáåÖ= íÜÉ= êÉäÉî~åí= áåÑçêã~íáçå= èìáÅâ= ~åÇ= é~áåäÉëëI= ëç= áí=Å~å=ÄÉ=ìëÉÇ=~ë=~å=ÉîÉêóÇ~ó=çåäáåÉ=êÉÑÉêÉåÅÉ=ÖìáÇÉK===

==

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Contents ====

1 krj_bo=pbqp= NKN= pÉí=fÇÉåíáíáÉë==1= NKO= pÉíë=çÑ=kìãÄÉêë==5= NKP= _~ëáÅ=fÇÉåíáíáÉë==7= NKQ= `çãéäÉñ=kìãÄÉêë==8= =2 ^idb_o^= OKN= c~ÅíçêáåÖ=cçêãìä~ë==12= OKO= mêçÇìÅí=cçêãìä~ë==13= OKP= mçïÉêë==14= OKQ= oççíë==15= OKR= içÖ~êáíÜãë==16= OKS= bèì~íáçåë==18= OKT= fåÉèì~äáíáÉë==19= OKU= `çãéçìåÇ=fåíÉêÉëí=cçêãìä~ë==22= =3 dbljbqov= PKN= oáÖÜí=qêá~åÖäÉ==24= PKO= fëçëÅÉäÉë=qêá~åÖäÉ==27= PKP= bèìáä~íÉê~ä=qêá~åÖäÉ==28= PKQ= pÅ~äÉåÉ=qêá~åÖäÉ==29= PKR= pèì~êÉ==33= PKS= oÉÅí~åÖäÉ==34= PKT= m~ê~ääÉäçÖê~ã==35= PKU= oÜçãÄìë==36= PKV= qê~éÉòçáÇ==37= PKNM= fëçëÅÉäÉë=qê~éÉòçáÇ==38= PKNN= fëçëÅÉäÉë=qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==40= PKNO= qê~éÉòçáÇ=ïáíÜ=fåëÅêáÄÉÇ=`áêÅäÉ==41=

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PKNP= háíÉ==42= PKNQ= `óÅäáÅ=nì~Çêáä~íÉê~ä==43= PKNR= q~åÖÉåíá~ä=nì~Çêáä~íÉê~ä==45= PKNS= dÉåÉê~ä=nì~Çêáä~íÉê~ä==46= PKNT= oÉÖìä~ê=eÉñ~Öçå==47= PKNU= oÉÖìä~ê=mçäóÖçå==48= PKNV= `áêÅäÉ==50= PKOM= pÉÅíçê=çÑ=~=`áêÅäÉ==53= PKON= pÉÖãÉåí=çÑ=~=`áêÅäÉ==54= PKOO= `ìÄÉ==55= PKOP= oÉÅí~åÖìä~ê=m~ê~ääÉäÉéáéÉÇ==56= PKOQ= mêáëã==57= PKOR= oÉÖìä~ê=qÉíê~ÜÉÇêçå==58= PKOS= oÉÖìä~ê=móê~ãáÇ==59= PKOT= cêìëíìã=çÑ=~=oÉÖìä~ê=móê~ãáÇ==61= PKOU= oÉÅí~åÖìä~ê=oáÖÜí=tÉÇÖÉ==62= PKOV= mä~íçåáÅ=pçäáÇë==63= PKPM= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê==66= PKPN= oáÖÜí=`áêÅìä~ê=`óäáåÇÉê=ïáíÜ=~å=lÄäáèìÉ=mä~åÉ=c~ÅÉ==68= PKPO= oáÖÜí=`áêÅìä~ê=`çåÉ==69= PKPP= cêìëíìã=çÑ=~=oáÖÜí=`áêÅìä~ê=`çåÉ==70= PKPQ= péÜÉêÉ==72= PKPR= péÜÉêáÅ~ä=`~é==72= PKPS= péÜÉêáÅ~ä=pÉÅíçê==73= PKPT= péÜÉêáÅ~ä=pÉÖãÉåí==74= PKPU= péÜÉêáÅ~ä=tÉÇÖÉ==75= PKPV= bääáéëçáÇ==76= PKQM= `áêÅìä~ê=qçêìë==78= = =4 qofdlkljbqov= QKN= o~Çá~å=~åÇ=aÉÖêÉÉ=jÉ~ëìêÉë=çÑ=^åÖäÉë==80= QKO= aÉÑáåáíáçåë=~åÇ=dê~éÜë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==81= QKP= páÖåë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==86= QKQ= qêáÖçåçãÉíêáÅ=cìåÅíáçåë=çÑ=`çããçå=^åÖäÉë==87= QKR= jçëí=fãéçêí~åí=cçêãìä~ë==88=

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QKS= oÉÇìÅíáçå=cçêãìä~ë==89= QKT= mÉêáçÇáÅáíó=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKU= oÉä~íáçåë=ÄÉíïÉÉå=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==90= QKV= ^ÇÇáíáçå=~åÇ=pìÄíê~Åíáçå=cçêãìä~ë==91= QKNM= açìÄäÉ=^åÖäÉ=cçêãìä~ë==92= QKNN= jìäíáéäÉ=^åÖäÉ=cçêãìä~ë==93= QKNO= e~äÑ=^åÖäÉ=cçêãìä~ë==94= QKNP= e~äÑ=^åÖäÉ=q~åÖÉåí=fÇÉåíáíáÉë==94= QKNQ= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=mêçÇìÅí==95= QKNR= qê~åëÑçêãáåÖ=çÑ=qêáÖçåçãÉíêáÅ=bñéêÉëëáçåë=íç=pìã==97=== QKNS= mçïÉêë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==98= QKNT= dê~éÜë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==99= QKNU= mêáåÅáé~ä=s~äìÉë=çÑ=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==102= QKNV= oÉä~íáçåë=ÄÉíïÉÉå=fåîÉêëÉ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==103= QKOM= qêáÖçåçãÉíêáÅ=bèì~íáçåë==106= QKON= oÉä~íáçåë=íç=eóéÉêÄçäáÅ=cìåÅíáçåë==106= = =5 j^qof`bp=^ka=abqbojfk^kqp= RKN= aÉíÉêãáå~åíë==107= RKO= mêçéÉêíáÉë=çÑ=aÉíÉêãáå~åíë==109= RKP= j~íêáÅÉë==110= RKQ= léÉê~íáçåë=ïáíÜ=j~íêáÅÉë==111= RKR= póëíÉãë=çÑ=iáåÉ~ê=bèì~íáçåë==114= = =6 sb`qlop= SKN= sÉÅíçê=`ççêÇáå~íÉë==118= SKO= sÉÅíçê=^ÇÇáíáçå==120= SKP= sÉÅíçê=pìÄíê~Åíáçå==122= SKQ= pÅ~äáåÖ=sÉÅíçêë==122= SKR= pÅ~ä~ê=mêçÇìÅí==123= SKS= sÉÅíçê=mêçÇìÅí==125= SKT= qêáéäÉ=mêçÇìÅí=127= = =7 ^k^ivqf`=dbljbqov= TKN= låÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==130=

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TKO= qïç=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==131= TKP= píê~áÖÜí=iáåÉ=áå=mä~åÉ==139= TKQ= `áêÅäÉ==149= TKR= bääáéëÉ==152= TKS= eóéÉêÄçä~==154= TKT= m~ê~Äçä~==158= TKU= qÜêÉÉ=-aáãÉåëáçå~ä=`ççêÇáå~íÉ=póëíÉã==161= TKV= mä~åÉ==165= TKNM= píê~áÖÜí=iáåÉ=áå=pé~ÅÉ==175= TKNN= nì~ÇêáÅ=pìêÑ~ÅÉë==180= TKNO= péÜÉêÉ==189= = =8 afccbobkqf^i=`^i`rirp= UKN= cìåÅíáçåë=~åÇ=qÜÉáê=dê~éÜë==191= UKO= iáãáíë=çÑ=cìåÅíáçåë==208= UKP= aÉÑáåáíáçå=~åÇ=mêçéÉêíáÉë=çÑ=íÜÉ=aÉêáî~íáîÉ==209= UKQ= q~ÄäÉ=çÑ=aÉêáî~íáîÉë==211= UKR= eáÖÜÉê=lêÇÉê=aÉêáî~íáîÉë==215= UKS= ^ééäáÅ~íáçåë=çÑ=aÉêáî~íáîÉ==217= UKT= aáÑÑÉêÉåíá~ä==221= UKU= jìäíáî~êá~ÄäÉ=cìåÅíáçåë==222= UKV= aáÑÑÉêÉåíá~ä=léÉê~íçêë==225= = =9 fkqbdo^i=`^i`rirp= VKN= fåÇÉÑáåáíÉ=fåíÉÖê~ä==227= VKO= fåíÉÖê~äë=çÑ=o~íáçå~ä=cìåÅíáçåë==228= VKP= fåíÉÖê~äë=çÑ=fêê~íáçå~ä=cìåÅíáçåë==231= VKQ= fåíÉÖê~äë=çÑ=qêáÖçåçãÉíêáÅ=cìåÅíáçåë==237= VKR= fåíÉÖê~äë=çÑ=eóéÉêÄçäáÅ=cìåÅíáçåë==241= VKS= fåíÉÖê~äë=çÑ=bñéçåÉåíá~ä=~åÇ=içÖ~êáíÜãáÅ=cìåÅíáçåë==242= VKT= oÉÇìÅíáçå=cçêãìä~ë==243= VKU= aÉÑáåáíÉ=fåíÉÖê~ä==247= VKV= fãéêçéÉê=fåíÉÖê~ä==253= VKNM= açìÄäÉ=fåíÉÖê~ä==257= VKNN= qêáéäÉ=fåíÉÖê~ä==269=

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VKNO= iáåÉ=fåíÉÖê~ä==275= VKNP= pìêÑ~ÅÉ=fåíÉÖê~ä==285= = =10 afccbobkqf^i=bnr^qflkp= NMKN= cáêëí=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==295= NMKO= pÉÅçåÇ=lêÇÉê=lêÇáå~êó=aáÑÑÉêÉåíá~ä=bèì~íáçåë==298= NMKP= pçãÉ=m~êíá~ä=aáÑÑÉêÉåíá~ä=bèì~íáçåë==302= = =11 pbofbp= NNKN= ^êáíÜãÉíáÅ=pÉêáÉë==304= NNKO= dÉçãÉíêáÅ=pÉêáÉë==305= NNKP= pçãÉ=cáåáíÉ=pÉêáÉë==305= NNKQ= fåÑáåáíÉ=pÉêáÉë==307= NNKR= mêçéÉêíáÉë=çÑ=`çåîÉêÖÉåí=pÉêáÉë==307= NNKS= `çåîÉêÖÉåÅÉ=qÉëíë==308= NNKT= ^äíÉêå~íáåÖ=pÉêáÉë==310= NNKU= mçïÉê=pÉêáÉë==311= NNKV= aáÑÑÉêÉåíá~íáçå=~åÇ=fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë==312= NNKNM= q~óäçê=~åÇ=j~Åä~ìêáå=pÉêáÉë==313= NNKNN= mçïÉê=pÉêáÉë=bñé~åëáçåë=Ñçê=pçãÉ=cìåÅíáçåë==314= NNKNO= _áåçãá~ä=pÉêáÉë==316= NNKNP= cçìêáÉê=pÉêáÉë==316= = =12 mol_^_fifqv= NOKN= mÉêãìí~íáçåë=~åÇ=`çãÄáå~íáçåë==318= NOKO= mêçÄ~Äáäáíó=cçêãìä~ë==319= = = =

==

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1

Chapter 1

Number Sets ====

1.1 Set Identities =pÉíëW=^I=_I=`=råáîÉêë~ä=ëÉíW=f=`çãéäÉãÉåí=W= ^′ =mêçéÉê=ëìÄëÉíW= _^⊂ ==bãéíó=ëÉíW=∅ =råáçå=çÑ=ëÉíëW= _^∪ =fåíÉêëÉÅíáçå=çÑ=ëÉíëW= _^∩ =aáÑÑÉêÉåÅÉ=çÑ=ëÉíëW= _y^ ==

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CHAPTER 1. NUMBER SETS

2

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Figure 1. =

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CHAPTER 1. NUMBER SETS

3

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Page 14: 1300 Math Formulas

CHAPTER 1. NUMBER SETS

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Figure 3. =

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5

1.2 Sets of Numbers =k~íìê~ä=åìãÄÉêëW=k=tÜçäÉ=åìãÄÉêëW= Mk =

fåíÉÖÉêëW=w=mçëáíáîÉ=áåíÉÖÉêëW= +w =kÉÖ~íáîÉ=áåíÉÖÉêëW= −w =o~íáçå~ä=åìãÄÉêëW=n=oÉ~ä=åìãÄÉêëW=o==`çãéäÉñ=åìãÄÉêëW=`====

26. k~íìê~ä=kìãÄÉêë `çìåíáåÖ=åìãÄÉêëW { }KIPIOINk = K=

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6

31. oÉ~ä=kìãÄÉêë==råáçå=çÑ=ê~íáçå~ä=~åÇ=áêê~íáçå~ä=åìãÄÉêëW=oK=

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Figure 5. ======

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7

1.3 Basic Identities =oÉ~ä=åìãÄÉêëW=~I=ÄI=Å=

==

34. ^ÇÇáíáîÉ=fÇÉåíáíó=~M~ =+ =

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8

43. ^ëëçÅá~íáîÉ=çÑ=jìäíáéäáÅ~íáçå=( ) ( )ÅÄ~ÅÄ~ ⋅⋅=⋅⋅

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1.4 Complex Numbers =k~íìê~ä=åìãÄÉêW=å=fã~Öáå~êó=ìåáíW=á=`çãéäÉñ=åìãÄÉêW=ò=oÉ~ä=é~êíW=~I=Å=fã~Öáå~êó=é~êíW=ÄáI=Çá=jçÇìäìë=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=êI= Nê I= Oê =^êÖìãÉåí=çÑ=~=ÅçãéäÉñ=åìãÄÉêW=ϕ I= Nϕ I= Oϕ =

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ááN = = ááR = = áá NåQ =+ =NáO −= = NáS −= = Ná OåQ −=+ =ááP −= = ááT −= = áá PåQ −=+ =

46.

NáQ = = NáU = = Ná åQ = ==47. Äá~ò += ==48. `çãéäÉñ=mä~åÉ=

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CHAPTER 1. NUMBER SETS

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55. mçä~ê=mêÉëÉåí~íáçå=çÑ=`çãéäÉñ=kìãÄÉêë=( )ϕ+ϕ=+ ëáåáÅçëêÄá~ =

=56. jçÇìäìë=~åÇ=^êÖìãÉåí=çÑ=~=`çãéäÉñ=kìãÄÉê=

fÑ= Äá~ + =áë=~=ÅçãéäÉñ=åìãÄÉêI=íÜÉå=OO Ä~ê += =EãçÇìäìëFI==

~

Ä~êÅí~å=ϕ =E~êÖìãÉåíFK=

=57. mêçÇìÅí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( )OOONNNON ëáåáÅçëêëáåáÅçëêòò ϕ+ϕ⋅ϕ+ϕ=⋅ =( ) ( )[ ]ONONON ëáåáÅçëêê ϕ+ϕ+ϕ+ϕ= =

=58. `çåàìÖ~íÉ=kìãÄÉêë=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ ëáåáÅçëêëáåáÅçëê|||||||||||||||||||||

==59. fåîÉêëÉ=çÑ=~=`çãéäÉñ=kìãÄÉê=áå=mçä~ê=oÉéêÉëÉåí~íáçå=

( ) ( ) ( )[ ]ϕ−+ϕ−=ϕ+ϕ

ëáåáÅçëê

N

ëáåáÅçëê

N=

Page 21: 1300 Math Formulas

CHAPTER 1. NUMBER SETS

11

60. nìçíáÉåí=áå=mçä~ê=oÉéêÉëÉåí~íáçå=( )( ) ( ) ( )[ ]ONON

O

N

OOO

NNN

O

N ëáåáÅçëê

ê

ëáåáÅçëê

ëáåáÅçëê

ò

òϕ−ϕ+ϕ−ϕ=

ϕ+ϕϕ+ϕ

= =

=61. mçïÉê=çÑ=~=`çãéäÉñ=kìãÄÉê=

( )[ ] ( ) ( )[ ]ϕ+ϕ=ϕ+ϕ= åëáåáåÅçëêëáåáÅçëêò ååå ==62. cçêãìä~=±aÉ=jçáîêÉ≤=

( ) ( ) ( )ϕ+ϕ=ϕ+ϕ åëáåáåÅçëëáåáÅçë å ==63. kíÜ=oççí=çÑ=~=`çãéäÉñ=kìãÄÉê=

( )

π+ϕ

+π+ϕ

=ϕ+ϕ=å

âOëáåá

å

âOÅçëêëáåáÅçëêò ååå I==

ïÜÉêÉ==NåIIOINIMâ −= K K==

=64. bìäÉê∞ë=cçêãìä~=

ñëáåáñÅçëÉáñ += ===

Page 22: 1300 Math Formulas

12

Chapter 2

Algebra ====

2.1 Factoring Formulas =oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==k~íìê~ä=åìãÄÉêW=å=

==

65. ( )( )Ä~Ä~Ä~ OO −+=− ==

66. ( )( )OOPP Ä~Ä~Ä~Ä~ ++−=− ==67. ( )( )OOPP Ä~Ä~Ä~Ä~ +−+=+ ==68. ( )( ) ( )( )( )OOOOOOQQ Ä~Ä~Ä~Ä~Ä~Ä~ ++−=+−=− ==

69. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ ++++−=− ==70. ( )( )QPOOPQRR Ä~ÄÄ~Ä~~Ä~Ä~ +−+−+=+ ==71. fÑ=å=áë=çÇÇI=íÜÉå=

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +−−+−+=+ K K===72. fÑ=å=áë=ÉîÉåI=íÜÉå==

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− +++++−=− K I==

Page 23: 1300 Math Formulas

CHAPTER 2. ALGEBRA

13

( )( )NåOåOPåOåNååå Ä~ÄÄ~Ä~~Ä~Ä~ −−−−− −+−+−+=+ K K====

2.2 Product Formulas oÉ~ä=åìãÄÉêëW=~I=ÄI=Å==tÜçäÉ=åìãÄÉêëW=åI=â===

73. ( ) OOO Ä~ÄO~Ä~ +−=− ==

74. ( ) OOO Ä~ÄO~Ä~ ++=+ ==

75. ( ) POOPP Ä~ÄPÄ~P~Ä~ −+−=− ==

76. ( ) POOPP Ä~ÄPÄ~P~Ä~ +++=+ ==

77. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ +−+−=− ==

78. ( ) QPOOPQQ Ä~ÄQÄ~SÄ~Q~Ä~ ++++=+ ==79. _áåçãá~ä=cçêãìä~=

( ) IÄ`~Ä`Ä~`Ä~`~`Ä~ åå

åNåNå

åOOåO

åNåN

ååM

åå +++++=+ −−

−− K

ïÜÉêÉ= ( )>âå>â

>å`â

å

−= =~êÉ=íÜÉ=Äáåçãá~ä=ÅçÉÑÑáÅáÉåíëK=

=

80. ( ) ÄÅO~ÅO~ÄOÅÄ~ÅÄ~ OOOO +++++=++ ==

81. ( ) ++++++=+++++ OOOOOO îìÅÄ~îìÅÄ~ KK =( )ìîÄîÄìÄÅ~î~ì~Å~ÄO +++++++++++ KKK =

Page 24: 1300 Math Formulas

CHAPTER 2. ALGEBRA

14

2.3 Powers =_~ëÉë=EéçëáíáîÉ=êÉ~ä=åìãÄÉêëFW=~I=Ä==mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

==

82. åãåã ~~~ += ==

83. åã

å

ã

~~

~ −= =

=

84. ( ) ããã Ä~~Ä = ==

85. ã

ãã

Ä

~

Ä

~=

=

=

86. ( ) ãååã ~~ = ==87. N~M = I= M~ ≠ ==88. N~N = ==

89. ã

ã

~

N~ =− =

=

90. å ãå

ã

~~ = ======

Page 25: 1300 Math Formulas

CHAPTER 2. ALGEBRA

15

2.4 Roots =_~ëÉëW=~I=Ä==mçïÉêë=Eê~íáçå~ä=åìãÄÉêëFW=åI=ã=

MÄI~ ≥ =Ñçê=ÉîÉå=êççíë=E âOå = I= kâ∈ F===

91. ååå Ä~~Ä = ==

92. åã åããå Ä~Ä~ = ==

93. å

å

å

Ä

~

Ä

~= I= MÄ ≠ =

=

94. åãå

ã

åã å

åã ã

ã

å

Ä

~

Ä

~

Ä

~== I= MÄ ≠ K=

=

95. ( ) å ãéé

å ã ~~ = ==

96. ( ) ~~åå = =

=

97. åé ãéå ã ~~ = ==

98. å

ãå ã ~~ = =

=

99. ãåã å ~~ = ==

100. ( ) å ããå ~~ = =

=

Page 26: 1300 Math Formulas

CHAPTER 2. ALGEBRA

16

101. ~

~

~

N å Nå

å

= I= M~ ≠ K=

=

102. O

Ä~~

O

Ä~~Ä~

OO −−±

−+=± =

=

103. Ä~

Ä~

Ä~

N

−=

±m

=

===

2.5 Logarithms =mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=ñI=óI=~I=ÅI=â=k~íìê~ä=åìãÄÉêW=å====

104. aÉÑáåáíáçå=çÑ=içÖ~êáíÜã=ñäçÖó ~= =áÑ=~åÇ=çåäó=áÑ= ó~ñ = I= M~ > I= N~ ≠ K=

=105. MNäçÖ~ = =

=106. N~äçÖ~ = =

=

107.

<∞+>∞−

=N~áÑ

N~áÑMäçÖ~ =

=108. ( ) óäçÖñäçÖñóäçÖ ~~~ += =

=

109. óäçÖñäçÖó

ñäçÖ ~~~ −= =

Page 27: 1300 Math Formulas

CHAPTER 2. ALGEBRA

17

110. ( ) ñäçÖåñäçÖ ~å

~ = =

=

111. ñäçÖå

NñäçÖ ~

å~ = =

=

112. ÅäçÖñäçÖ~äçÖ

ñäçÖñäçÖ ~Å

Å

Å~ ⋅== I= MÅ > I= NÅ ≠ K=

=

113. ~äçÖ

NÅäçÖ

Å~ = =

=114. ñäçÖ~~ñ = =

=115. içÖ~êáíÜã=íç=_~ëÉ=NM=

ñäçÖñäçÖNM = =

=116. k~íìê~ä=içÖ~êáíÜã=

ñäåñäçÖÉ = I==

ïÜÉêÉ= KTNUOUNUOUKOâ

NNäáãÉ

â

â=

+=

∞→=

=

117. ñäåQPQOVQKMñäåNMäå

NñäçÖ == =

=

118. ñäçÖPMORURKOñäçÖÉäçÖ

Nñäå == =

=====

Page 28: 1300 Math Formulas

CHAPTER 2. ALGEBRA

18

2.6 Equations =oÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI=éI=èI=ìI=î=pçäìíáçåëW= Nñ I= Oñ I= Nó I= Oó I= Pó =

==

119. iáåÉ~ê=bèì~íáçå=áå=låÉ=s~êá~ÄäÉ=

MÄ~ñ =+ I=~

Äñ −= K==

=120. nì~Çê~íáÅ=bèì~íáçå=

MÅÄñ~ñO =++ I=~O

~ÅQÄÄñ

O

OIN

−±−= K=

=121. aáëÅêáãáå~åí=

~ÅQÄa O −= ==

122. sáÉíÉ∞ë=cçêãìä~ë=

fÑ= MèéññO =++ I=íÜÉå==

=−=+

èññ

éññ

ON

ON K=

=

123. MÄñ~ñO =+ I= MñN = I=~

ÄñO −= K=

=

124. MÅ~ñ O =+ I=~

Åñ OIN −±= K=

=125. `ìÄáÅ=bèì~íáçåK=`~êÇ~åç∞ë=cçêãìä~K==

MèéóóP =++ I==

Page 29: 1300 Math Formulas

CHAPTER 2. ALGEBRA

19

îìóN += I= ( ) ( ) áîìO

Pîì

O

Nó PIO +±+−= I==

ïÜÉêÉ==

P

OO

P

é

O

è

O

èì

+

+−= I= P

OO

P

é

O

è

O

èî

+

−−= K==

==

2.7 Inequalities s~êá~ÄäÉëW=ñI=óI=ò=

oÉ~ä=åìãÄÉêëW=

åPON ~II~I~I~

ÇIÅIÄI~

KI=ãI=å=

aÉíÉêãáå~åíëW=aI= ña I= óa I= òa ==

==

126. fåÉèì~äáíáÉëI=fåíÉêî~ä=kçí~íáçåë=~åÇ=dê~éÜë===

fåÉèì~äáíó= fåíÉêî~ä=kçí~íáçå= dê~éÜ=Äñ~ ≤≤ = [ ]ÄI~ =

=Äñ~ ≤< = ( ]ÄI~ =

=Äñ~ <≤ = [ )ÄI~ =

=Äñ~ << = ( )ÄI~ =

=Äñ ≤<∞− I=

Äñ ≤ =( ]ÄI∞− =

=Äñ <<∞− I=

Äñ < =( )ÄI∞− =

=∞<≤ ñ~ I=

~ñ ≥ =[ )∞I~ =

=∞<< ñ~ I=

~ñ > =( )∞I~ =

=

Page 30: 1300 Math Formulas

CHAPTER 2. ALGEBRA

20

127. fÑ= Ä~ > I=íÜÉå= ~Ä < K==128. fÑ= Ä~ > I=íÜÉå= MÄ~ >− =çê= M~Ä <− K==129. fÑ= Ä~ > I=íÜÉå= ÅÄÅ~ +>+ K==130. fÑ= Ä~ > I=íÜÉå= ÅÄÅ~ −>− K==131. fÑ= Ä~ > =~åÇ= ÇÅ > I=íÜÉå= ÇÄÅ~ +>+ K==132. fÑ= Ä~ > =~åÇ= ÇÅ > I=íÜÉå= ÅÄÇ~ −>− K==133. fÑ= Ä~ > =~åÇ= Mã > I=íÜÉå= ãÄã~ > K==

134. fÑ= Ä~ > =~åÇ= Mã > I=íÜÉå=ã

Ä

ã

~> K=

=135. fÑ= Ä~ > =~åÇ= Mã < I=íÜÉå= ãÄã~ < K==

136. fÑ= Ä~ > =~åÇ= Mã < I=íÜÉå=ã

Ä

ã

~< K=

=137. fÑ= Ä~M << =~åÇ= Må > I=íÜÉå= åå Ä~ < K==138. fÑ= Ä~M << =~åÇ= Må < I=íÜÉå= åå Ä~ > K==

139. fÑ= Ä~M << I=íÜÉå= åå Ä~ < K==

140. O

Ä~~Ä

+≤ I==

ïÜÉêÉ= M~ > =I= MÄ > X=~å=Éèì~äáíó=áë=î~äáÇ=çåäó=áÑ= Ä~ = K===

141. O~

N~ ≥+ I=ïÜÉêÉ= M~ > X=~å=Éèì~äáíó=í~âÉë=éä~ÅÉ=çåäó=~í= N~ = K=

Page 31: 1300 Math Formulas

CHAPTER 2. ALGEBRA

21

142. å

~~~~~~ åONå

åON

+++≤

KK I=ïÜÉêÉ= M~II~I~ åON >K K=

=

143. fÑ= MÄ~ñ >+ =~åÇ= M~ > I=íÜÉå=~

Äñ −> K=

=

144. fÑ= MÄ~ñ >+ =~åÇ= M~ < I=íÜÉå=~

Äñ −< K==

=145. MÅÄñ~ñ O >++ ==

= M~ > = M~ < ====

Ma> =

=

=

Nññ < I= Oññ > =

=

==

ON ñññ << =

===

Ma= =

=

ññN < I= Nññ > =

==∅∈ñ =

===

Ma< =

=

=∞<<∞− ñ =

===∅∈ñ =

=

Page 32: 1300 Math Formulas

CHAPTER 2. ALGEBRA

22

146. Ä~Ä~ +≤+ =

=147. fÑ= ~ñ < I=íÜÉå= ~ñ~ <<− I=ïÜÉêÉ= M~ > K=

=148. fÑ= ~ñ > I=íÜÉå= ~ñ −< =~åÇ= ~ñ > I=ïÜÉêÉ= M~ > K=

=

149. fÑ= ~ñO < I=íÜÉå= ~ñ < I=ïÜÉêÉ= M~ > K=

=

150. fÑ= ~ñO > I=íÜÉå= ~ñ > I=ïÜÉêÉ= M~ > K=

=

151. fÑ=( )( ) MñÖ

ñÑ> I=íÜÉå=

( ) ( )( )

≠>⋅

MñÖ

MñÖñÑK=

=

152. ( )( ) MñÖ

ñÑ< I=íÜÉå=

( ) ( )( )

≠<⋅

MñÖ

MñÖñÑK=

===

2.8 Compound Interest Formulas =cìíìêÉ=î~äìÉW=^=fåáíá~ä=ÇÉéçëáíW=`=^ååì~ä=ê~íÉ=çÑ=áåíÉêÉëíW=ê=kìãÄÉê=çÑ=óÉ~êë=áåîÉëíÉÇW=í=kìãÄÉê=çÑ=íáãÉë=ÅçãéçìåÇÉÇ=éÉê=óÉ~êW=å=

==

153. dÉåÉê~ä=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=åí

å

êN`^

+= =

=

Page 33: 1300 Math Formulas

CHAPTER 2. ALGEBRA

23

154. páãéäáÑáÉÇ=`çãéçìåÇ=fåíÉêÉëí=cçêãìä~=fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=çåÅÉ=éÉê=óÉ~êI=íÜÉå=íÜÉ=éêÉîáçìë=Ñçêãìä~=ëáãéäáÑáÉë=íçW=

( )íêN`^ += K==155. `çåíáåìçìë=`çãéçìåÇ=fåíÉêÉëí=

fÑ=áåíÉêÉëí=áë=ÅçãéçìåÇÉÇ=Åçåíáåì~ääó=E ∞→å FI=íÜÉå==êí`É^ = K=

==

Page 34: 1300 Math Formulas

24

Chapter 3

Geometry ====

3.1 Right Triangle =iÉÖë=çÑ=~=êáÖÜí=íêá~åÖäÉW=~I=Ä=eóéçíÉåìëÉW=Å=^äíáíìÇÉW=Ü=jÉÇá~åëW= ~ã I= Äã I= Åã =

^åÖäÉëW=α Iβ =o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=^êÉ~W=p===

==

Figure 8. =

156. °=β+α VM ==

Page 35: 1300 Math Formulas

CHAPTER 3. GEOMETRY

25

157. β==α ÅçëÅ

~ëáå =

=

158. β==α ëáåÅ

ÄÅçë =

=

159. β==α ÅçíÄ

~í~å =

=

160. β==α í~å~

ÄÅçí =

=

161. β==α ÉÅÅçëÄ

ÅëÉÅ =

=

162. β==α ëÉÅ~

ÅÉÅÅçë =

=163. móíÜ~ÖçêÉ~å=qÜÉçêÉã=

OOO ÅÄ~ =+ ==

164. ÑÅ~O = I= ÖÅÄO = I==

ïÜÉêÉ= Ñ= ~åÇ= Å= ~êÉ= éêçàÉÅíáçåë= çÑ= íÜÉ= äÉÖë= ~= ~åÇ= ÄI= êÉëéÉÅ-íáîÉäóI=çåíç=íÜÉ=ÜóéçíÉåìëÉ=ÅK==

===== ==

Figure 9. =

Page 36: 1300 Math Formulas

CHAPTER 3. GEOMETRY

26

165. ÑÖÜO = I===ïÜÉêÉ=Ü=áë=íÜÉ=~äíáíìÇÉ=Ñêçã=íÜÉ=êáÖÜí=~åÖäÉK==

=

166. Q

~Äã

OOO

~ −= I=Q

Ä~ã

OOO

Ä −= I===

ïÜÉêÉ= ~ã =~åÇ= Äã =~êÉ=íÜÉ=ãÉÇá~åë=íç=íÜÉ=äÉÖë=~=~åÇ=ÄK==

=

==

Figure 10. =

167. O

ÅãÅ = I==

ïÜÉêÉ= Åã =áë=íÜÉ=ãÉÇá~å=íç=íÜÉ=ÜóéçíÉåìëÉ=ÅK=

=

168. ÅãO

Åo == =

=

169. ÅÄ~

O

ÅÄ~ê

++=

−+= =

=170. ÅÜ~Ä = =

==

Page 37: 1300 Math Formulas

CHAPTER 3. GEOMETRY

27

171. O

ÅÜ

O

~Äp == =

===

3.2 Isosceles Triangle =_~ëÉW=~=iÉÖëW=Ä=_~ëÉ=~åÖäÉW=β =sÉêíÉñ=~åÖäÉW=α =^äíáíìÇÉ=íç=íÜÉ=Ä~ëÉW=Ü=mÉêáãÉíÉêW=i=^êÉ~W=p=

==

==

Figure 11. =

172. O

VMα

−°=β =

=

173. Q

~ÄÜ

OOO −= =

Page 38: 1300 Math Formulas

CHAPTER 3. GEOMETRY

28

174. ÄO~i += ==

175. α== ëáåO

Ä

O

~Üp

O

=

===

3.3 Equilateral Triangle =páÇÉ=çÑ=~=Éèìáä~íÉê~ä=íêá~åÖäÉW=~=^äíáíìÇÉW=Ü=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===

==

Figure 12. =

176. O

P~Ü = =

=

Page 39: 1300 Math Formulas

CHAPTER 3. GEOMETRY

29

177. P

P~Ü

P

Oo == =

=

178. O

o

S

P~Ü

P

Nê === =

=179. ~Pi = =

=

180. Q

P~

O

~Üp

O

== =

===

3.4 Scalene Triangle E^=íêá~åÖäÉ=ïáíÜ=åç=íïç=ëáÇÉë=Éèì~äF=

==páÇÉë=çÑ=~=íêá~åÖäÉW=~I=ÄI=Å=

pÉãáéÉêáãÉíÉêW=O

ÅÄ~é

++= ==

^åÖäÉë=çÑ=~=íêá~åÖäÉW= γβα II =^äíáíìÇÉë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ÅÄ~ ÜIÜIÜ =

jÉÇá~åë=íç=íÜÉ=ëáÇÉë=~I=ÄI=ÅW= ÅÄ~ ãIãIã =

_áëÉÅíçêë=çÑ=íÜÉ=~åÖäÉë= γβα II W= ÅÄ~ íIíIí =

o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=^êÉ~W=p===

Page 40: 1300 Math Formulas

CHAPTER 3. GEOMETRY

30

===== ==

Figure 13. =

181. °=γ+β+α NUM ==

182. ÅÄ~ >+ I==~ÅÄ >+ I==ÄÅ~ >+ K=

=183. ÅÄ~ <− I==

~ÅÄ <− I==

ÄÅ~ <− K=

=184. jáÇäáåÉ=

O

~è = I= ~ööè K=

=

===== ==

Figure 14. =

Page 41: 1300 Math Formulas

CHAPTER 3. GEOMETRY

31

185. i~ï=çÑ=`çëáåÉë=

α−+= ÅçëÄÅOÅÄ~ OOO I=

β−+= Åçë~ÅOÅ~Ä OOO I=

γ−+= Åçë~ÄOÄ~Å OOO K==

186. i~ï=çÑ=páåÉë=

oOëáå

Å

ëáå

Ä

ëáå

~=

γ=

β=

αI==

ïÜÉêÉ=o=áë=íÜÉ=ê~Çáìë=çÑ=íÜÉ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉK===

187. pQ

~ÄÅ

ÜO

ÜO

ÜO

ÄÅ

ëáåO

Å

ëáåO

Ä

ëáåO

~o

ÅÄ~

====γ

= =

=

188. ( )( )( )é

ÅéÄé~éêO −−−= I==

ÅÄ~ Ü

N

Ü

N

Ü

N

ê

N++= K=

=

189. ( )( )ÄÅ

ÅéÄé

Oëáå

−−=

αI=

( )ÄÅ

~éé

OÅçë

−=

αI=

( )( )( )~éé

ÅéÄé

Oí~å

−−−

K=

=

190. ( )( )( )ÅéÄé~éé~

OÜ~ −−−= I=

( )( )( )ÅéÄé~ééÄ

OÜÄ −−−= I=

( )( )( )ÅéÄé~ééÅ

OÜÅ −−−= K=

Page 42: 1300 Math Formulas

CHAPTER 3. GEOMETRY

32

191. β=γ= ëáåÅëáåÄÜ~ I=

α=γ= ëáåÅëáå~ÜÄ I=

α=β= ëáåÄëáå~ÜÅ K=

=

192. Q

~

O

ÅÄã

OOOO~ −

+= I==

Q

Ä

O

Å~ã

OOOOÄ −

+= I==

Q

Å

O

Ä~ã

OOOOÅ −

+= K=

=

===== ==

Figure 15. =

193. ~ãP

O^j = I= Äã

P

O_j = I= Åã

P

O`j = =EcáÖKNRFK=

=

194. ( )( )O

O~

ÅÄ

~éÄÅéQí

+−

= I==

( )( )O

Å~

Äé~ÅéQí

+−

= I==

( )( )O

Ä~

Åé~ÄéQí

+−

= K=

=

Page 43: 1300 Math Formulas

CHAPTER 3. GEOMETRY

33

195. O

ÅÜ

O

ÄÜ

O

~Üp ÅÄ~ === I==

O

ëáåÄÅ

O

ëáå~Å

O

ëáå~Äp

α=

β=

γ= I==

( )( )( )ÅéÄé~éép −−−= =EeÉêçå∞ë=cçêãìä~FI=

éêp = I==

oQ

~ÄÅp = I=

γβα= ëáåëáåëáåoOp O I=

Oí~å

Oí~å

Oí~åép O γβα

= K=

===

3.5 Square páÇÉ=çÑ=~=ëèì~êÉW=~=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p==

==

Figure 16.

Page 44: 1300 Math Formulas

CHAPTER 3. GEOMETRY

34

196. O~Ç = ===

197. O

O~

O

Ço == =

=

198. O

~ê = =

=199. ~Qi = =

=

200. O~p = ====

3.6 Rectangle =páÇÉë=çÑ=~=êÉÅí~åÖäÉW=~I=Ä=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=^êÉ~W=p===

==

Figure 17. =

201. OO Ä~Ç += ==

Page 45: 1300 Math Formulas

CHAPTER 3. GEOMETRY

35

202. O

Ço = =

=203. ( )Ä~Oi += =

=204. ~Äp = =

===

3.7 Parallelogram =páÇÉë=çÑ=~=é~ê~ääÉäçÖê~ãW=~I=Ä=aá~Öçå~äëW= ON ÇIÇ =`çåëÉÅìíáîÉ=~åÖäÉëW= βαI =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =^äíáíìÇÉW=Ü==mÉêáãÉíÉêW=i=^êÉ~W=p===

===== ==

Figure 18. =

205. °=β+α NUM ==

206. ( )OOOO

ON Ä~OÇÇ +=+ =

=

Page 46: 1300 Math Formulas

CHAPTER 3. GEOMETRY

36

207. β=α= ëáåÄëáåÄÜ ==

208. ( )Ä~Oi += ==

209. α== ëáå~Ä~Üp I==

ϕ= ëáåÇÇO

Np ON K=

===

3.8 Rhombus =páÇÉ=çÑ=~=êÜçãÄìëW=~=aá~Öçå~äëW= ON ÇIÇ =`çåëÉÅìíáîÉ=~åÖäÉëW= βαI =^äíáíìÇÉW=e=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===

===== ==

Figure 19. =

Page 47: 1300 Math Formulas

CHAPTER 3. GEOMETRY

37

210. °=β+α NUM ==

211. OOO

ON ~QÇÇ =+ =

=

212. ~O

ÇÇëáå~Ü ON=α= =

=

213. O

ëáå~

~Q

ÇÇ

O

Üê ON α

=== =

=214. ~Qi = =

=

215. α== ëáå~~Üp O I==

ONÇÇO

Np = K=

===

3.9 Trapezoid =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=^êÉ~W=p===

Page 48: 1300 Math Formulas

CHAPTER 3. GEOMETRY

38

==

Figure 20. =

216. O

Ä~è

+= =

=

217. èÜÜO

Ä~p =⋅

+= =

===

3.10 Isosceles Trapezoid =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=iÉÖW=Å=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äW=Ç=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=^êÉ~W=p===

Page 49: 1300 Math Formulas

CHAPTER 3. GEOMETRY

39

==

Figure 21. =

218. O

Ä~è

+= =

=

219. OÅ~ÄÇ += ==

220. ( )OO ~ÄQ

NÅÜ −−= =

=

221. ( )( )Ä~ÅOÄ~ÅO

Å~ÄÅo

O

−++−+

= =

=

222. èÜÜO

Ä~p =⋅

+= =

======

Page 50: 1300 Math Formulas

CHAPTER 3. GEOMETRY

40

3.11 Isosceles Trapezoid with Inscribed Circle

=_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=iÉÖW=Å=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äW=Ç=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=o=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=^êÉ~W=p===

==

Figure 22. =

223. ÅOÄ~ =+ ==

224. ÅO

Ä~è =

+= =

=225. OOO ÅÜÇ += =

=

Page 51: 1300 Math Formulas

CHAPTER 3. GEOMETRY

41

226. O

O

Üê == =

=

227. ~

ÄS

Ä

~

U

Ä~ÅÜ

ÜO

Å

ÅN

O

Å

êQ

ÅÇ

ÜO

ÅÇo OO

O

+++

=+=+=== =

=228. ( ) ÅQÄ~Oi =+= =

=

229. ( )O

iêÅÜèÜ

O

~ÄÄ~Ü

O

Ä~p ===

+=⋅

+= ==

===

3.12 Trapezoid with Inscribed Circle =_~ëÉë=çÑ=~=íê~éÉòçáÇW=~I=Ä=i~íÉê~ä=ëáÇÉëW=ÅI=Ç=jáÇäáåÉW=è=^äíáíìÇÉW=Ü=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=^êÉ~W=p==

Page 52: 1300 Math Formulas

CHAPTER 3. GEOMETRY

42

==

Figure 23. =

230. ÇÅÄ~ +=+ ==

231. O

ÇÅ

O

Ä~è

+=

+= =

=232. ( ) ( )ÇÅOÄ~Oi +=+= =

=

233. èÜÜO

ÇÅÜ

O

Ä~p =⋅

+=⋅

+= I==

ϕ= ëáåÇÇO

Np ON K=

===

3.13 Kite =páÇÉë=çÑ=~=âáíÉW=~I=Ä=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉëW= γβα II =mÉêáãÉíÉêW=i=^êÉ~W=p===

Page 53: 1300 Math Formulas

CHAPTER 3. GEOMETRY

43

==

Figure 24. =

234. °=γ+β+α PSMO ==

235. ( )Ä~Oi += ==

236. O

ÇÇp ON= =

===

3.14 Cyclic Quadrilateral páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =fåíÉêå~ä=~åÖäÉëW= δγβα III =o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p=

Page 54: 1300 Math Formulas

CHAPTER 3. GEOMETRY

44

==

Figure 25. =

237. °=δ+β=γ+α NUM ==

238. míçäÉãó∞ë=qÜÉçêÉã=

ONÇÇÄÇ~Å =+ ==

239. ÇÅÄ~i +++= ==

240. ( )( )( )( )( )( )( )ÇéÅéÄé~é

ÅÇ~ÄÄÅ~ÇÄÇ~Å

Q

No

−−−−+++

= I==

ïÜÉêÉ=O

ié = K=

=

241. ϕ= ëáåÇÇO

Np ON I==

( )( )( )( )ÇéÅéÄé~ép −−−−= I==

ïÜÉêÉ=O

ié = K=

===

Page 55: 1300 Math Formulas

CHAPTER 3. GEOMETRY

45

3.15 Tangential Quadrilateral =páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===

==

Figure 26. =

242. ÇÄÅ~ +=+ ==

243. ( ) ( )ÇÄOÅ~OÇÅÄ~i +=+=+++= ==

244. ( ) ( )

éO

éÄ~Ä~ÇÇê

OOOO

ON −+−−

= I==

ïÜÉêÉ=O

ié = K==

=

Page 56: 1300 Math Formulas

CHAPTER 3. GEOMETRY

46

245. ϕ== ëáåÇÇO

Néêp ON =

===

3.16 General Quadrilateral =páÇÉë=çÑ=~=èì~Çêáä~íÉê~äW=~I=ÄI=ÅI=Ç=aá~Öçå~äëW= ON ÇIÇ =^åÖäÉ=ÄÉíïÉÉå=íÜÉ=Çá~Öçå~äëW=ϕ =fåíÉêå~ä=~åÖäÉëW= δγβα III =mÉêáãÉíÉêW=i=^êÉ~W=p===

======= ==

Figure 27. =

246. °=δ+γ+β+α PSM ==

247. ÇÅÄ~i +++= ==

Page 57: 1300 Math Formulas

CHAPTER 3. GEOMETRY

47

248. ϕ= ëáåÇÇO

Np ON =

===

3.17 Regular Hexagon =páÇÉW=~=fåíÉêå~ä=~åÖäÉW=α =pä~åí=ÜÉáÖÜíW=ã=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===

==

Figure 28. =

249. °=α NOM ==

250. O

P~ãê == =

Page 58: 1300 Math Formulas

CHAPTER 3. GEOMETRY

48

251. ~o = ==

252. ~Si = ==

253. O

PP~éêp

O

== I==

ïÜÉêÉ=O

ié = K=

===

3.18 Regular Polygon =páÇÉW=~=kìãÄÉê=çÑ=ëáÇÉëW=å=fåíÉêå~ä=~åÖäÉW=α =pä~åí=ÜÉáÖÜíW=ã=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=mÉêáãÉíÉêW=i=pÉãáéÉêáãÉíÉêW=é==^êÉ~W=p===

Page 59: 1300 Math Formulas

CHAPTER 3. GEOMETRY

49

==

Figure 29. =

254. °⋅−

=α NUMO

Oå=

=

255. °⋅−

=α NUMO

Oå=

=

256.

åëáåO

~o

π= =

=

257. Q

~o

åí~åO

~ãê

OO −=

π== =

=258. å~i = =

=

259. å

Oëáå

O

åop

O π= I==

Q

~oééêp

OO −== I==

Page 60: 1300 Math Formulas

CHAPTER 3. GEOMETRY

50

ïÜÉêÉ=O

ié = K==

===

3.19 Circle =o~ÇáìëW=o=aá~ãÉíÉêW=Ç=`ÜçêÇW=~=pÉÅ~åí=ëÉÖãÉåíëW=ÉI=Ñ=q~åÖÉåí=ëÉÖãÉåíW=Ö=`Éåíê~ä=~åÖäÉW=α =fåëÅêáÄÉÇ=~åÖäÉW=β =mÉêáãÉíÉêW=i=^êÉ~W=p===

260. O

ëáåoO~α

= =

=

==

Figure 30. =

Page 61: 1300 Math Formulas

CHAPTER 3. GEOMETRY

51

261. ONON ÄÄ~~ = ==

==

Figure 31. =

262. NN ÑÑÉÉ = ==

===== ==

Figure 32. =

263. NO ÑÑÖ = =

=

Page 62: 1300 Math Formulas

CHAPTER 3. GEOMETRY

52

===== ==

Figure 33. =

264. O

α=β =

=

==

Figure 34. =

265. ÇoOi π=π= ==

266. O

io

Q

Çop

OO =

π=π= ==

=

Page 63: 1300 Math Formulas

CHAPTER 3. GEOMETRY

53

3.20 Sector of a Circle =o~Çáìë=çÑ=~=ÅáêÅäÉW=o=^êÅ=äÉåÖíÜW=ë=`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α =mÉêáãÉíÉêW=i=^êÉ~W=p===

==

Figure 35. =

267. oñë = ==

268. °απ

=NUM

oë =

=269. oOëi += =

=

270. °απ

===PSM

o

O

ño

O

oëp

OO

==

==

Page 64: 1300 Math Formulas

CHAPTER 3. GEOMETRY

54

3.21 Segment of a Circle =o~Çáìë=çÑ=~=ÅáêÅäÉW=o=^êÅ=äÉåÖíÜW=ë=`ÜçêÇW=~=`Éåíê~ä=~åÖäÉ=Eáå=ê~Çá~åëFW=ñ=`Éåíê~ä=~åÖäÉ=Eáå=ÇÉÖêÉÉëFW=α =eÉáÖÜí=çÑ=íÜÉ=ëÉÖãÉåíW=Ü=mÉêáãÉíÉêW=i=^êÉ~W=p===

==

Figure 36. =

271. OÜÜoOO~ −= ==

272. OO ~oQO

NoÜ −−= I= oÜ < =

=273. ~ëi += =

=

Page 65: 1300 Math Formulas

CHAPTER 3. GEOMETRY

55

274. ( )[ ] ( )ñëáåñO

oëáå

NUMO

oÜo~ëo

O

Np

OO

−=

α−

°απ

=−−= I==

Ü~P

Op ≈ K=

===

3.22 Cube =bÇÖÉW=~==aá~Öçå~äW=Ç=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ëéÜÉêÉW=ê=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

=== ==

Figure 37. =

275. P~Ç = ==

276. O

~ê = =

=

Page 66: 1300 Math Formulas

CHAPTER 3. GEOMETRY

56

277. O

P~o = =

=278. O~Sp = =

=279. P~s = ==

===

3.23 Rectangular Parallelepiped =bÇÖÉëW=~I=ÄI=Å==aá~Öçå~äW=Ç=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

===== ==

Figure 38. =

280. OOO ÅÄ~Ç ++= ==

281. ( )ÄÅ~Å~ÄOp ++= ==

282. ~ÄÅs = ==

Page 67: 1300 Math Formulas

CHAPTER 3. GEOMETRY

57

3.24 Prism =i~íÉê~ä=ÉÇÖÉW=ä=eÉáÖÜíW=Ü=i~íÉê~ä=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

===== ==

Figure 39. =

283. _i pOpp += K===

284. i~íÉê~ä=^êÉ~=çÑ=~=oáÖÜí=mêáëã=( )ä~~~~p åPONi ++++= K =

=285. i~íÉê~ä=^êÉ~=çÑ=~å=lÄäáèìÉ=mêáëã=

éäpi = I==ïÜÉêÉ=é=áë=íÜÉ=éÉêáãÉíÉê=çÑ=íÜÉ=Åêçëë=ëÉÅíáçåK=

=

Page 68: 1300 Math Formulas

CHAPTER 3. GEOMETRY

58

286. Üps _= ==

287. `~î~äáÉêáDë=mêáåÅáéäÉ==dáîÉå=íïç=ëçäáÇë= áåÅäìÇÉÇ=ÄÉíïÉÉå=é~ê~ääÉä=éä~åÉëK=fÑ=ÉîÉêó=éä~åÉ=Åêçëë=ëÉÅíáçå=é~ê~ääÉä=íç=íÜÉ=ÖáîÉå=éä~åÉë=Ü~ë=íÜÉ=ë~ãÉ=~êÉ~=áå=ÄçíÜ=ëçäáÇëI=íÜÉå=íÜÉ=îçäìãÉë=çÑ=íÜÉ=ëçäáÇë=~êÉ=Éèì~äK====

3.25 Regular Tetrahedron =qêá~åÖäÉ=ëáÇÉ=äÉåÖíÜW=~=eÉáÖÜíW=Ü=^êÉ~=çÑ=Ä~ëÉW= _p =pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

==

Figure 40. =

288. ~P

OÜ = =

=

Page 69: 1300 Math Formulas

CHAPTER 3. GEOMETRY

59

289. Q

~Pp

O

_ = =

=

290. O~Pp = ==

291. OS

~Üp

P

Ns

P

_ == K==

===

3.26 Regular Pyramid =páÇÉ=çÑ=Ä~ëÉW=~=i~íÉê~ä=ÉÇÖÉW=Ä=eÉáÖÜíW=Ü=pä~åí=ÜÉáÖÜíW=ã==kìãÄÉê=çÑ=ëáÇÉëW=å==pÉãáéÉêáãÉíÉê=çÑ=Ä~ëÉW=é=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ëéÜÉêÉ=çÑ=Ä~ëÉW=ê=^êÉ~=çÑ=Ä~ëÉW= _p =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

Page 70: 1300 Math Formulas

CHAPTER 3. GEOMETRY

60

==

Figure 41. =

292. Q

~Äã

OO −= =

=

293.

åëáåO

ëáåÄQÜ

OOO

π

−π

= =

=

294. éã~ÄQå~Q

Nå~ã

O

Np OO

i =−== =

=295. éêp_ = =

=296. i_ ppp += =

=

297. éêÜP

NÜp

P

Ns _ == ==

===

Page 71: 1300 Math Formulas

CHAPTER 3. GEOMETRY

61

3.27 Frustum of a Regular Pyramid =

_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=

åPON

åPON

ÄIIÄIÄIÄ

~II~I~I~

K

K=

eÉáÖÜíW=Ü=pä~åí=ÜÉáÖÜíW=ã==^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =mÉêáãÉíÉê=çÑ=Ä~ëÉëW= Nm I= Om =pÅ~äÉ=Ñ~ÅíçêW=â=qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

==

Figure 42. =

298. â~

Ä

~

Ä

~

Ä

~

Ä

~

Ä

å

å

P

P

O

O

N

N ====== K =

=

Page 72: 1300 Math Formulas

CHAPTER 3. GEOMETRY

62

299. O

N

O âp

p= =

=

300. ( )O

mmãp ON

i

+= =

=301. ONi pppp ++= =

=

302. ( )OONN ppppP

Üs ++= =

=

303. [ ]ON

O

N ââNP

Üp

~

Ä

~

ÄN

P

Üps ++=

++= =

===

3.28 Rectangular Right Wedge =páÇÉë=çÑ=Ä~ëÉW=~I=Ä=qçé=ÉÇÖÉW=Å=eÉáÖÜíW=Ü=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

Page 73: 1300 Math Formulas

CHAPTER 3. GEOMETRY

63

==

Figure 43. =

304. ( ) ( )OOOOi Å~ÜÄÄÜQÅ~

O

Np −++++= =

=305. ~Äp_ = =

=306. i_ ppp += =

=

307. ( )Å~OS

ÄÜs += =

===

3.29 Platonic Solids =bÇÖÉW=~=o~Çáìë=çÑ=áåëÅêáÄÉÇ=ÅáêÅäÉW=ê=o~Çáìë=çÑ=ÅáêÅìãëÅêáÄÉÇ=ÅáêÅäÉW=o=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

Page 74: 1300 Math Formulas

CHAPTER 3. GEOMETRY

64

308. cáîÉ=mä~íçåáÅ=pçäáÇë=qÜÉ= éä~íçåáÅ= ëçäáÇë= ~êÉ= ÅçåîÉñ= éçäóÜÉÇê~= ïáíÜ= Éèìáî~äÉåí=Ñ~ÅÉë=ÅçãéçëÉÇ=çÑ=ÅçåÖêìÉåí=ÅçåîÉñ=êÉÖìä~ê=éçäóÖçåëK==

=pçäáÇ= kìãÄÉê=

çÑ=sÉêíáÅÉëkìãÄÉê=çÑ=bÇÖÉë=

kìãÄÉê=çÑ=c~ÅÉë=

pÉÅíáçå=

qÉíê~ÜÉÇêçå== Q= S= Q= PKOR=`ìÄÉ= U= NO= S= PKOO=

lÅí~ÜÉÇêçå= S= NO= U= PKOT=fÅçë~ÜÉÇêçå= NO= PM= OM= PKOT=

açÇÉÅ~ÜÉÇêçå= OM= PM= NO= PKOT===

Octahedron =

==

Figure 44. =

309. S

S~ê = =

=

310. O

O~o = =

=

Page 75: 1300 Math Formulas

CHAPTER 3. GEOMETRY

65

311. P~Op O= ==

312. P

O~s

P

= =

==

Icosahedron =

==

Figure 45. =

313. ( )NO

RPP~ê

+= =

=

314. ( )RROQ

~o += =

=

315. P~Rp O= ==

316. ( )NO

RP~Rs

P += =

==

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CHAPTER 3. GEOMETRY

66

Dodecahedron =

==

Figure 46. =

317. ( )O

RNNORNM~ê

+= =

=

318. ( )Q

RNP~o

+= =

=

319. ( )RORR~Pp O += =

=

320. ( )Q

RTNR~s

P += =

===

3.30 Right Circular Cylinder =o~Çáìë=çÑ=Ä~ëÉW=o=aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=

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CHAPTER 3. GEOMETRY

67

eÉáÖÜíW=e=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

===== ==

Figure 47. =

321. oeOpi π= ==

322. ( )

+π=+π=+=

O

ÇeÇoeoOpOpp _i =

=

323. eoeps O_ π== =

===

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CHAPTER 3. GEOMETRY

68

3.31 Right Circular Cylinder with an Oblique Plane Face

=o~Çáìë=çÑ=Ä~ëÉW=o=qÜÉ=ÖêÉ~íÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= NÜ =qÜÉ=ëÜçêíÉëí=ÜÉáÖÜí=çÑ=~=ëáÇÉW= OÜ =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

==

Figure 48. =

324. ( )ONi ÜÜop +π= ==

325. O

ONOO_ O

ÜÜooop

+π+π= =

=

Page 79: 1300 Math Formulas

CHAPTER 3. GEOMETRY

69

326.

++++π=+=O

ONOON_i O

ÜÜooÜÜoppp =

=

327. ( )ON

O

ÜÜO

os +

π= =

===

3.32 Right Circular Cone o~Çáìë=çÑ=Ä~ëÉW=o=aá~ãÉíÉê=çÑ=Ä~ëÉW=Ç=eÉáÖÜíW=e=pä~åí=ÜÉáÖÜíW=ã=i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =^êÉ~=çÑ=Ä~ëÉW= _p =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

==

Figure 49.

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CHAPTER 3. GEOMETRY

70

328. OO oãe −= ==

329. O

ãÇoãpi

π=π= =

=

330. O_ op π= =

=

331. ( )

+π=+π=+=

O

ÇãÇ

O

Noãoppp _i =

=

332. eoP

Nep

P

Ns O

_ π== =

===

3.33 Frustum of a Right Circular Cone =o~Çáìë=çÑ=Ä~ëÉëW=oI=ê=eÉáÖÜíW=e=pä~åí=ÜÉáÖÜíW=ã=pÅ~äÉ=Ñ~ÅíçêW=â=^êÉ~=çÑ=Ä~ëÉëW= Np I= Op =i~íÉê~ä=ëìêÑ~ÅÉ=~êÉ~W= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

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CHAPTER 3. GEOMETRY

71

==

Figure 50. =

333. ( )OO êoãe −−= =

=

334. âê

o= =

=

335. O

O

O

N

O âê

o

p

p== =

=336. ( )êoãpi +π= ==337. ( )[ ]êoãêopppp OO

iON +++π=++= ==

338. ( )OONN ppppP

Üs ++= =

=

339. [ ]ON

O

N ââNP

Üp

ê

o

ê

oN

P

Üps ++=

++= =

===

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CHAPTER 3. GEOMETRY

72

3.34 Sphere =o~ÇáìëW=o=aá~ãÉíÉêW=Ç=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==

==

Figure 51. =

340. OoQp π= ==

341. poP

S

Neo

P

Qs PP =π=π= =

===

3.35 Spherical Cap o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉW=ê=eÉáÖÜíW=Ü=^êÉ~=çÑ=éä~åÉ=Ñ~ÅÉW= _p =^êÉ~=çÑ=ëéÜÉêáÅ~ä=Å~éW= `p =

qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s=

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CHAPTER 3. GEOMETRY

73

==

Figure 52. =

342. ÜO

Üêo

OO += =

=343. O

_ êp π= ==344. ( )OO

` êÜp +π= =

=345. ( ) ( )OOO

`_ êoÜOêOÜppp +π=+π=+= =

=

346. ( ) ( )OOO ÜêPÜS

ÜoPÜS

s +π

=−π

= =

===

3.36 Spherical Sector =o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉ=çÑ=ëéÜÉêáÅ~ä=Å~éW=ê=eÉáÖÜíW=Ü=qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==

Page 84: 1300 Math Formulas

CHAPTER 3. GEOMETRY

74

====== === ==

Figure 53. =

347. ( )êÜOop +π= ==

348. ÜoP

Os Oπ= =

=kçíÉW=qÜÉ=ÖáîÉå= Ñçêãìä~ë=~êÉ=ÅçêêÉÅí=ÄçíÜ= Ñçê=±çéÉå≤= ~åÇ=±ÅäçëÉÇ≤=ëéÜÉêáÅ~ä=ëÉÅíçêK====

3.37 Spherical Segment =o~Çáìë=çÑ=ëéÜÉêÉW=o=o~Çáìë=çÑ=Ä~ëÉëW= Nê I= Oê =eÉáÖÜíW=Ü=^êÉ~=çÑ=ëéÜÉêáÅ~ä=ëìêÑ~ÅÉW= pp =

^êÉ~=çÑ=éä~åÉ=ÉåÇ=Ñ~ÅÉëW= Np I= Op =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==

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CHAPTER 3. GEOMETRY

75

===== ==

Figure 54. =

349. oÜOpp π= =

=350. ( )O

OO

NONp êêoÜOpppp ++π=++= =

=

351. ( )OOO

ON ÜêPêPÜ

S

Ns ++π= =

===

3.38 Spherical Wedge =o~ÇáìëW=o=aáÜÉÇê~ä=~åÖäÉ=áå=ÇÉÖêÉÉëW=ñ=aáÜÉÇê~ä=~åÖäÉ=áå=ê~Çá~åëW=α =^êÉ~=çÑ=ëéÜÉêáÅ~ä=äìåÉW= ip =qçí~ä=ëìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

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CHAPTER 3. GEOMETRY

76

==

Figure 55. =

352. ñoOVM

op O

O

i =απ

= =

=

353. ñoOoVM

oop OO

OO +π=α

π+π= =

=

354. ñoP

O

OTM

os P

P

=απ

= =

===

3.39 Ellipsoid =pÉãá-~ñÉëW=~I=ÄI=Å=sçäìãÉW=s=

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CHAPTER 3. GEOMETRY

77

======= ==

Figure 56. =

355. ~ÄÅP

Qs π= =

===

Prolate Spheroid =pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä~ > F=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

356.

+π=

É

É~êÅëáå~ÄÄOp I==

ïÜÉêÉ=~

Ä~É

OO −= K=

=

357. ~ÄP

Qs Oπ= =

=

Page 88: 1300 Math Formulas

CHAPTER 3. GEOMETRY

78

Oblate Spheroid =pÉãá-~ñÉëW=~I=ÄI=Ä=E Ä~ < F=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s===

358.

+π=~LÄÉ

~

ÄÉ~êÅëáåÜ~

ÄÄOp I==

ïÜÉêÉ=Ä

~ÄÉ

OO −= K=

=

359. ~ÄP

Qs Oπ= =

===

3.40 Circular Torus =j~àçê=ê~ÇáìëW=o=jáåçê=ê~ÇáìëW=ê=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉW=s==

Page 89: 1300 Math Formulas

CHAPTER 3. GEOMETRY

79

== =Picture 57.

=360. oêQp Oπ= ==361. OOoêOs π= =

==

Page 90: 1300 Math Formulas

80

Chapter 4

Trigonometry ====^åÖäÉëW=α I=β =oÉ~ä=åìãÄÉêë=EÅççêÇáå~íÉë=çÑ=~=éçáåíFW=ñI=ó==tÜçäÉ=åìãÄÉêW=â===

4.1 Radian and Degree Measures of Angles =

362. ?QRDNTRTNUM

ê~ÇN °≈π°

= =

=

363. ê~ÇMNTQRPKMê~ÇNUM

N ≈π

=° =

=

364. ê~ÇMMMOVNKMê~ÇSMNUM

DN ≈⋅π

= =

=

365. ê~ÇMMMMMRKMê~ÇPSMMNUM

?N ≈⋅π

= =

=366. ==

^åÖäÉ=EÇÉÖêÉÉëF=

M= PM= QR= SM= VM= NUM= OTM= PSM=

^åÖäÉ=Eê~Çá~åëF= M=

S

π=

Q

π=

P

π=

O

π= π =

O

Pπ= πO =

===

Page 91: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

81

4.2 Definitions and Graphs of Trigonometric Functions

=

= ==

Figure 58. =

367. ê

óëáå =α =

=

368. ê

ñÅçë =α =

=

369. ñ

óí~å =α =

=

370. ó

ñÅçí =α =

=

Page 92: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

82

371. ñ

êëÉÅ =α =

=

372. ó

êÅçëÉÅ =α =

=373. páåÉ=cìåÅíáçå=

ñëáåó = I= NñëáåN ≤≤− K==

=

Figure 59. =

374. `çëáåÉ=cìåÅíáçå==ñÅçëó = I= NñÅçëN ≤≤− K=

Page 93: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

83

==

Figure 60. =

375. q~åÖÉåí=cìåÅíáçå=

ñí~åó = I= ( )O

NâOñπ

+≠ I= Kñí~å ∞≤≤∞− =

=

==

Figure 61. =

Page 94: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

84

376. `çí~åÖÉåí=cìåÅíáçå==ñÅçíó = I= π≠ âñ I== ∞≤≤∞− ñÅçí K=

=

==

Figure 62. =

377. pÉÅ~åí=cìåÅíáçå=

ñëÉÅó = I= ( )O

NâOñπ

+≠ K=

==

Page 95: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

85

==

Figure 63. =

378. `çëÉÅ~åí=cìåÅíáçå==ñÉÅÅçëó = I= π≠ âñ K=

=

Figure 64.

Page 96: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

86

4.3. Signs of Trigonometric Functions 379. ==

nì~Çê~åí=páåα =

`çëα =

q~åα =

`çíα =

pÉÅα =

`çëÉÅ=α =

f= H= H= H= H= H= H=ff= H= �= �= �= �= H=fff= �= �= H= H= �= �=fs= �= H= �= �= H= �==

==

380. ==

=

Figure 65.

==========

Page 97: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

87

4.4 Trigonometric Functions of Common Angles 381. =°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí αëÉÅ = αÅçëÉÅ =

M= M= M= N= M= ∞ = N= ∞ =

PM=S

π=

O

N=

O

P=

P

N= P =

P

O= O=

QR=Q

π=

O

O=

O

O= N= N= O = O =

SM=P

π=

O

P=

O

N= P =

P

N= O=

P

O=

VM=O

π= N= M= ∞ = M= ∞ = N=

NOM=P

Oπ=

O

P=

O

N− = P− =

P

N− � O− =

P

O=

NUM= π = M= N− = M= ∞ = N− = ∞ =

OTM=O

Pπ= N− = M= ∞ = M= ∞ = N− =

PSM= πO = M= N= M= ∞ = N= ∞ ==============

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CHAPTER 4. TRIGONOMETRY

88

382. =°α = ê~Çα = αëáå = αÅçë = αí~å = αÅçí =

NR=NO

π=

Q

OS −=

Q

OS += PO− = PO+ =

NU=NM

π=

Q

NR −=

Q

RONM+R

ROR−= ROR+ =

PS=R

π=

Q

RONM−Q

NR +=

NR

RONM

+−

RONM

NR

+

=

RQ=NM

Pπ=

Q

NR +=

Q

RONM−RONM

NR

+NR

RONM

+−

=

TO=R

Oπ=

Q

RONM+Q

NR −= ROR+ = R

ROR−

=

TR=NO

Rπ=

Q

OS +=

Q

OS −= PO+ = PO− =

===

4.5 Most Important Formulas =

383. NÅçëëáå OO =α+α ==

384. Ní~åëÉÅ OO =α−α ==

385. NÅçíÅëÅ OO =α−α ==

386. αα

=αÅçë

ëáåí~å =

Page 99: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

89

387. αα

=αëáå

ÅçëÅçí =

=388. NÅçíí~å =α⋅α =

=

389. α

=αÅçë

NëÉÅ =

=

390. α

=αëáå

NÅçëÉÅ =

===

4.6 Reduction Formulas =

391. ==

β = βëáå = βÅçë = βí~å = βÅçí =

α− = α− ëáå = α+ Åçë = α− í~å = α− Åçí =

α−°VM = α+ Åçë = α+ ëáå = α+ Åçí = α+ í~å =

α+°VM = α+ Åçë = α− ëáå = α− Åçí = α− í~å =

α−°NUM α+ ëáå = α− Åçë = α− í~å = α− Åçí =

α+°NUM α− ëáå = α− Åçë = α+ í~å = α+ Åçí =

α−°OTM α− Åçë = α− ëáå = α+ Åçí = α+ í~å =

α+°OTM α− Åçë = α+ ëáå = α− Åçí = α− í~å =

α−°PSM α− ëáå = α+ Åçë = α− í~å = α− Åçí =

α+°PSM α+ ëáå = α+ Åçë = α+ í~å = α+ Åçí ==

=====

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CHAPTER 4. TRIGONOMETRY

90

4.7 Periodicity of Trigonometric Functions =

392. ( ) α=π±α ëáååOëáå I=éÉêáçÇ= πO =çê= °PSM K==

393. ( ) α=π±α ÅçëåOÅçë I=éÉêáçÇ= πO =çê= °PSM K==

394. ( ) α=π±α í~ååí~å I=éÉêáçÇ=π =çê= °NUM K==

395. ( ) α=π±α ÅçíåÅçí I=éÉêáçÇ=π =çê= °NUM K====

4.8 Relations between Trigonometric Functions

=

396. ( ) NQO

ÅçëOOÅçëNO

NÅçëNëáå OO −

π

−α

=α−±=α−±=α =

=

Oí~åN

Oí~åO

O α+

α

= =

=

397. ( ) NO

ÅçëOOÅçëNO

NëáåNÅçë OO −

α=α+±=α−±=α =

=

Oí~åN

Oí~åN

O

O

α+

α−

= =

=

398. αα−

=α+

α=−α±=

αα

=αOëáå

OÅçëN

OÅçëN

OëáåNëÉÅ

Åçë

ëáåí~å O =

Page 101: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

91

=

Oí~åN

Oí~åO

OÅçëN

OÅçëN

O α+

α

=α+α−

±= =

=

399. α−

α=

αα+

=−α±=αα

=αOÅçëN

Oëáå

Oëáå

OÅçëNNÅëÅ

ëáå

ÅçëÅçí O =

=

Oí~åO

Oí~åN

OÅçëN

OÅçëNO

α

α−

=α−α+

±= =

=

400.

Oí~åN

Oí~åN

í~åNÅçë

NëÉÅ

O

O

O

α−

α+

=α+±=α

=α =

=

401.

Oí~åO

Oí~åN

ÅçíNëáå

NÅëÅ

O

O

α

α+

=α+±=α

=α =

===

4.9 Addition and Subtraction Formulas =

402. ( ) αβ+βα=β+α ÅçëëáåÅçëëáåëáå ==

403. ( ) αβ−βα=−α ÅçëëáåÅçëëáåóëáå ==

404. ( ) βα−βα=β+α ëáåëáåÅçëÅçëÅçë ==

405. ( ) βα+βα=β−α ëáåëáåÅçëÅçëÅçë =

Page 102: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

92

406. ( )βα−β+α

=β+αí~åí~åN

í~åí~åí~å =

=

407. ( )βα+β−α

=β−αí~åí~åN

í~åí~åí~å =

=

408. ( )β+αβα−

=β+αí~åí~å

í~åí~åNÅçí =

=

409. ( )β−αβα+

=β−αí~åí~å

í~åí~åNÅçí =

===

4.10 Double Angle Formulas =

410. α⋅α=α ÅçëëáåOOëáå ==

411. NÅçëOëáåONëáåÅçëOÅçë OOOO −α=α−=α−α=α ==

412. α−α

=α−α

=αí~åÅçí

O

í~åN

í~åOOí~å

O=

=

413. O

í~åÅçí

ÅçíO

NÅçíOÅçí

O α−α=

α−α

=α =

======

Page 103: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

93

4.11 Multiple Angle Formulas =

414. α−α⋅α=α−α=α POP ëáåëáåÅçëPëáåQëáåPPëáå ==

415. α⋅α−α⋅α=α ÅçëëáåUÅçëëáåQQëáå P ==

416. α+α−α=α RP ëáåNSëáåOMëáåRRëáå ==

417. α⋅α−α=α−α=α OPP ëáåÅçëPÅçëÅçëPÅçëQPÅçë ==

418. NÅçëUÅçëUQÅçë OQ +α−α=α ==

419. α+α−α=α ÅçëRÅçëOMÅçëNSRÅçë PR ==

420. α−α−α

=αO

P

í~åPN

í~åí~åPPí~å =

=

421. α+α−α−α

=αQO

P

í~åí~åSN

í~åQí~åQQí~å =

=

422. α+α−α+α−α

=αQO

PR

í~åRí~åNMN

í~åRí~åNMí~åRí~å =

=

423. NÅçíP

ÅçíPÅçíPÅçí

O

P

−αα−α

=α =

=

424. α−αα+α−

=αP

QO

í~åQí~åQ

í~åí~åSNQÅçí ==

=

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CHAPTER 4. TRIGONOMETRY

94

425. α+α−α

α+α−=α

í~åRí~åNMí~å

í~åRí~åNMNRÅçí

PR

QO

=

===

4.12 Half Angle Formulas =

426. O

ÅçëN

Oëáå

α−±=

α=

=

427. O

ÅçëN

OÅçë

α+±=

α=

=

428. α−α=αα−

=α+

α=

α+α−

±=α

ÅçíÅëÅëáå

ÅçëN

ÅçëN

ëáå

ÅçëN

ÅçëN

Oí~å =

=

429. α+α=αα+

=α−

α=

α−α+

±=α

ÅçíÅëÅëáå

ÅçëN

ÅçëN

ëáå

ÅçëN

ÅçëN

OÅçí =

===

4.13 Half Angle Tangent Identities =

430.

Oí~åN

Oí~åO

ëáåO α+

α

=α =

=

Page 105: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

95

431.

Oí~åN

Oí~åN

ÅçëO

O

α+

α−

=α =

=

432.

Oí~åN

Oí~åO

í~åO α−

α

=α =

=

433.

Oí~åO

Oí~åN

Åçí

O

α

α−

=α =

===

4.14 Transforming of Trigonometric Expressions to Product

=

434. O

ÅçëO

ëáåOëáåëáåβ−αβ+α

=β+α =

=

435. O

ëáåO

ÅçëOëáåëáåβ−αβ+α

=β−α =

=

436. O

ÅçëO

ÅçëOÅçëÅçëβ−αβ+α

=β+α =

=

437. O

ëáåO

ëáåOÅçëÅçëβ−αβ+α

−=β−α =

=

Page 106: 1300 Math Formulas

CHAPTER 4. TRIGONOMETRY

96

438. ( )β⋅α

β+α=β+α

ÅçëÅçë

ëáåí~åí~å =

=

439. ( )β⋅α

β−α=β−α

ÅçëÅçë

ëáåí~åí~å =

=

440. ( )β⋅α

α+β=β+α

ëáåëáå

ëáåÅçíÅçí =

=

441. ( )β⋅α

α−β=β−α

ëáåëáå

ëáåÅçíÅçí =

=

442.

α+π

=

α−π

=α+αQ

ëáåOQ

ÅçëOëáåÅçë =

=

443.

α+π

=

α−π

=α−αQ

ÅçëOQ

ëáåOëáåÅçë =

=

444. ( )β⋅α

β−α=β+α

ëáåÅçë

ÅçëÅçíí~å =

=

445. ( )β⋅α

β+α−=β−α

ëáåÅçë

ÅçëÅçíí~å =

=

446. O

ÅçëOÅçëN O α=α+ =

=

447. O

ëáåOÅçëN O α=α− =

=

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CHAPTER 4. TRIGONOMETRY

97

448.

α

−π

=α+OQ

ÅçëOëáåN O =

=

449.

α

−π

=α−OQ

ëáåOëáåN O =

===

4.15 Transforming of Trigonometric Expressions to Sum

=

450. ( ) ( )O

ÅçëÅçëëáåëáå

β+α−β−α=β⋅α =

=

451. ( ) ( )O

ÅçëÅçëÅçëÅçë

β+α+β−α=β⋅α =

=

452. ( ) ( )O

ëáåëáåÅçëëáå

β+α+β−α=β⋅α =

=

453. β+αβ+α

=β⋅αÅçíÅçí

í~åí~åí~åí~å =

=

454. β+αβ+α

=β⋅αí~åí~å

ÅçíÅçíÅçíÅçí =

=

455. β+αβ+α

=β⋅αí~åÅçí

Åçíí~åÅçíí~å =

===

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CHAPTER 4. TRIGONOMETRY

98

4.16 Powers of Trigonometric Functions =

456. O

OÅçëNëáåO α−

=α =

=

457. Q

PëáåëáåPëáåP α−α

=α =

=

458. U

POÅçëQQÅçëëáåQ +α−α

=α =

=

459. NS

RëáåPëáåRëáåNMëáåR α+α−α

=α =

=

460. PO

SÅçëQÅçëSOÅçëNRNMëáåS α−α+α−

=α =

=

461. O

OÅçëNÅçëO α+

=α =

=

462. Q

PÅçëÅçëPÅçëP α+α

=α =

=

463. U

POÅçëQQÅçëÅçëQ +α+α

=α =

=

464. NS

RÅçëPëáåRÅçëNMÅçëR α+α+α

=α =

=

465. PO

SÅçëQÅçëSOÅçëNRNMÅçëS α+α+α+

=α =

=

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CHAPTER 4. TRIGONOMETRY

99

4.17 Graphs of Inverse Trigonometric Functions

=466. fåîÉêëÉ=páåÉ=cìåÅíáçå==

ñ~êÅëáåó = I= NñN ≤≤− I=O

ñ~êÅëáåO

π≤≤

π− K=

=

==

Figure 66. =

467. fåîÉêëÉ=`çëáåÉ=cìåÅíáçå==ñ~êÅÅçëó = I= NñN ≤≤− I= π≤≤ ñ~êÅÅçëM K=

=

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CHAPTER 4. TRIGONOMETRY

100

==

Figure 67. =

468. fåîÉêëÉ=q~åÖÉåí=cìåÅíáçå==

ñ~êÅí~åó = I= ∞≤≤∞− ñ I=O

ñ~êÅí~åO

π<<

π− K=

=

===== ==

Figure 68.

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CHAPTER 4. TRIGONOMETRY

101

469. fåîÉêëÉ=`çí~åÖÉåí=cìåÅíáçå==ñÅçí~êÅó = I= ∞≤≤∞− ñ I= π<< ñÅçí~êÅM K=

===== =

Figure 69. =

470. fåîÉêëÉ=pÉÅ~åí=cìåÅíáçå==

( ] [ ) KIOO

IMñëÉÅ~êÅIINNIñIñ=~êÅëÉÅó

ππ

π

∈∞∪−∞−∈=

=

Figure 70.

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CHAPTER 4. TRIGONOMETRY

102

471. fåîÉêëÉ=`çëÉÅ~åí=cìåÅíáçå==

( ] [ ) KO

IMMIO

ñÅëÅ~êÅIINNIñIñ~êÅÅëÅó

π

π−∈∞∪−∞−∈=

==

Figure 71. ==

4.18 Principal Values of Inverse Trigonometric Functions 472. ñ = M=

O

N=

O

O=

O

PN=

ñ~êÅëáå = °M = °PM = °QR = °SM °VMñ~êÅÅçë = °VM °SM = °QR = °PM °M =

ñ = O

N−

O

O−

O

P− N− = =

ñ~êÅëáå =°−PM

=°− QR °− SM

°− VM=

=

ñ~êÅÅçë =°NOM

=°NPR = °NRM =

°NUM=

=

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CHAPTER 4. TRIGONOMETRY

103

473. ñ = M=

P

PN= P =

P

P− N− = P− =

ñ~êÅí~å = °M = °PM °QR °SM °−PM°− QR

=°− SM =

ñÅçí~êÅ = °VM °SM °QR °PM °NOM =°NPR

=°NRM =

===

4.19 Relations between Inverse Trigonometric Functions

=474. ( ) ñ~êÅëáåñ~êÅëáå −=− =

=

475. ñ~êÅÅçëO

ñ~êÅëáå −π

= =

=

476. OñN~êÅÅçëñ~êÅëáå −= I= NñM ≤≤ K==

477. OñN~êÅÅçëñ~êÅëáå −−= I= MñN ≤≤− K==

478. OñN

ñ~êÅí~åñ~êÅëáå

−= I= NñO < K=

=

479. ñ

ñNÅçí~êÅñ~êÅëáå

O−= I= NñM ≤< K=

=

480. π−−

ñNÅçí~êÅñ~êÅëáå

O

I= MñN <≤− K=

=481. ( ) ñ~êÅÅçëñ~êÅÅçë −π=− =

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CHAPTER 4. TRIGONOMETRY

104

482. ñ~êÅëáåO

ñ~êÅÅçë −π

= =

=

483. OñN~êÅëáåñ~êÅÅçë −= I= NñM ≤≤ K==

484. OñN~êÅëáåñ~êÅÅçë −−π= I= MñN ≤≤− K==

485. ñ

ñN~êÅí~åñ~êÅÅçë

O−= I= NñM ≤< K=

=

486. ñ

ñN~êÅí~åñ~êÅÅçë

O−+π= I= MñN <≤− K=

=

487. OñN

ñÅçí~êÅñ~êÅÅçë

−= I= NñN ≤≤− K=

=488. ( ) ñ~êÅí~åñ~êÅí~å −=− =

=

489. ñÅçí~êÅO

ñ~êÅí~å −π

= =

=

490. OñN

ñ~êÅëáåñ~êÅí~å

+= =

=

491. OñN

N~êÅÅçëñ~êÅí~å

+= I= Mñ ≥ K=

=

492. OñN

N~êÅÅçëñ~êÅí~å

+−= I= Mñ ≤ K=

=

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CHAPTER 4. TRIGONOMETRY

105

493. ñ

N~êÅí~å

Oñ~êÅí~å −

π= I= Mñ > K=

=

494. ñ

N~êÅí~å

Oñ~êÅí~å −

π−= I= Mñ < K=

=

495. ñ

NÅçí~êÅñ~êÅí~å = I= Mñ > K=

=

496. π−=ñ

NÅçí~êÅñ~êÅí~å I= Mñ < K=

=497. ( ) ñÅçí~êÅñÅçí~êÅ −π=− =

=

498. ñ~êÅí~åO

ñÅçí~êÅ −π

= =

=

499. OñN

N~êÅëáåñÅçí~êÅ

+= I= Mñ > K=

=

500. OñN

N~êÅëáåñÅçí~êÅ

+−π= I= Mñ < K=

=

501. OñN

ñ~êÅÅçëñÅçí~êÅ

+= =

=

502. ñ

N~êÅí~åñÅçí~êÅ = I= Mñ > K=

=

503. ñ

N~êÅí~åñÅçí~êÅ +π= I= Mñ < K=

==

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CHAPTER 4. TRIGONOMETRY

106

4.20 Trigonometric Equations =tÜçäÉ=åìãÄÉêW=å===

504. ~ñëáå = I= ( ) å~~êÅëáåNñ å π+−= ==

505. ~ñÅçë = I= åO~~êÅÅçëñ π+±= ==

506. ~ñí~å = I= å~~êÅí~åñ π+= ==

507. ~ñÅçí = I= å~Åçí~êÅñ π+= ====

4.21 Relations to Hyperbolic Functions =fã~Öáå~êó=ìåáíW=á===

508. ( ) ñëáåÜááñëáå = ==

509. ( ) ñí~åÜááñí~å = ==

510. ( ) ñÅçíÜááñÅçí −= ==

511. ( ) ñëÉÅÜáñëÉÅ = ==

512. ( ) ñÅëÅÜááñÅëÅ −= ====

Page 117: 1300 Math Formulas

107

Chapter 5

Matrices and Determinants ====j~íêáÅÉëW=^I=_I=`=bäÉãÉåíë=çÑ=~=ã~íêáñW= á~ I= áÄ I= áà~ I= áàÄ I= áàÅ =

aÉíÉêãáå~åí=çÑ=~=ã~íêáñW= ^ÇÉí =jáåçê=çÑ=~å=ÉäÉãÉåí= áà~ W= áàj =

`çÑ~Åíçê=çÑ=~å=ÉäÉãÉåí= áà~ W= áà` =

qê~åëéçëÉ=çÑ=~=ã~íêáñW= q^ I= ^ú

=^Çàçáåí=çÑ=~=ã~íêáñW= ^~Çà =qê~ÅÉ=çÑ=~=ã~íêáñW= ^íê =

fåîÉêëÉ=çÑ=~=ã~íêáñW= N^− =oÉ~ä=åìãÄÉêW=â=oÉ~ä=î~êá~ÄäÉëW= áñ =k~íìê~ä=åìãÄÉêëW=ãI=å=====

5.1 Determinants =

513. pÉÅçåÇ=lêÇÉê=aÉíÉêãáå~åí=

NOONOO

NN Ä~Ä~Ä~

Ä~^ÇÉí −== =

=====

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CHAPTER 5. MATRICES AND DETERMINANTS

108

514. qÜáêÇ=lêÇÉê=aÉíÉêãáå~åí=

−++== POONNPPNOPNOPPOONN

PPPOPN

OPOOON

NPNONN

~~~~~~~~~

~~~

~~~

~~~

^ÇÉí =

PNOONPPPONNOPOOPNN ~~~~~~~~~ −−− =

=515. p~êêìë=oìäÉ=E^êêçï=oìäÉF=

==

Figure 72. =

516. k-íÜ=lêÇÉê=aÉíÉêãáå~åí=

åååàOåNå

áåáàOáNá

åOàOOOON

åNàNNONN

~~~~

~~~~

~~~~

~~~~

^ÇÉí

KK

KKKKKK

KK

KKKKKK

KK

KK

= =

=517. jáåçê=

qÜÉ=ãáåçê= áàj =~ëëçÅá~íÉÇ=ïáíÜ=íÜÉ=ÉäÉãÉåí= áà~ =çÑ=å-íÜ=çêÇÉê=

ã~íêáñ= ^= áë= íÜÉ= ( )Nå− -íÜ= çêÇÉê= ÇÉíÉêãáå~åí= ÇÉêáîÉÇ= Ñêçã=

íÜÉ=ã~íêáñ=^=Äó=ÇÉäÉíáçå=çÑ=áíë=á-íÜ=êçï=~åÇ=à-íÜ=ÅçäìãåK====

Page 119: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

109

518. `çÑ~Åíçê=

( ) áààá

áà jN` +−= =

=519. i~éä~ÅÉ=bñé~åëáçå=çÑ=å-íÜ=lêÇÉê=aÉíÉêãáå~åí=

i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=á-íÜ=êçï=

∑=

Nàáàáà`~^ÇÉí I= åIIOINá K= K=

i~éä~ÅÉ=Éñé~åëáçå=Äó=ÉäÉãÉåíë=çÑ=íÜÉ=à-íÜ=Åçäìãå=

∑=

Nááàáà`~^ÇÉí I= åIIOINà K= K==

===

5.2 Properties of Determinants =

520. qÜÉ==î~äìÉ==çÑ=~=ÇÉíÉêãáå~åí=êÉã~áåë==ìåÅÜ~åÖÉÇ=áÑ=êçïë=~êÉ=ÅÜ~åÖÉÇ=íç=Åçäìãåë=~åÇ=Åçäìãåë=íç=êçïëK=

=OO

NN

ON

ON

Ä~

Ä~

ÄÄ

~~= ==

=521. fÑ=íïç==êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áåíÉêÅÜ~åÖÉÇI=íÜÉ=ëáÖå=çÑ=

íÜÉ=ÇÉíÉêãáå~åí=áë=ÅÜ~åÖÉÇK=

NN

OO

OO

NN

Ä~

Ä~

Ä~

Ä~−= =

=522. fÑ=íïç=êçïë==Eçê=íïç=ÅçäìãåëF=~êÉ==áÇÉåíáÅ~äI=íÜÉ=î~äìÉ=çÑ=íÜÉ=

ÇÉíÉêãáå~åí=áë=òÉêçK=

M~~

~~

OO

NN = =

=

Page 120: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

110

523. fÑ==íÜÉ===ÉäÉãÉåíë==çÑ==~åó=êçï==Eçê=ÅçäìãåF=~êÉ=ãìäíáéäáÉÇ=Äó=====~==Åçããçå==Ñ~ÅíçêI==íÜÉ==ÇÉíÉêãáå~åí==áë==ãìäíáéäáÉÇ==Äó==íÜ~í=Ñ~ÅíçêK=

OO

NN

OO

NN

Ä~

Ä~â

Ä~

âÄâ~= =

=524. fÑ==íÜÉ==ÉäÉãÉåíë==çÑ==~åó==êçï==Eçê==ÅçäìãåF=~êÉ=áåÅêÉ~ëÉÇ=Eçê=

ÇÉÅêÉ~ëÉÇFÄó=Éèì~ä=ãìäíáéäÉë=çÑ=íÜÉ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=çÑ=~åó=çíÜÉê=êçï==Eçê=ÅçäìãåFI==íÜÉ=î~äìÉ=çÑ=íÜÉ=ÇÉíÉêãáå~åí=áë=ìåÅÜ~åÖÉÇK=

OO

NN

OOO

NNN

Ä~

Ä~

ÄâÄ~

ÄâÄ~=

++

=

===

5.3 Matrices =

525. aÉÑáåáíáçå=^å= åã× =ã~íêáñ=^=áë=~=êÉÅí~åÖìä~ê=~êê~ó=çÑ=ÉäÉãÉåíë=Eåìã-ÄÉêë=çê=ÑìåÅíáçåëF=ïáíÜ=ã=êçïë=~åÇ=å=ÅçäìãåëK==

[ ]

==

ãåOãNã

åOOOON

åNNONN

áà

~~~

~~~

~~~

~^

K

MMM

K

K

==

=526. pèì~êÉ=ã~íêáñ=áë=~=ã~íêáñ=çÑ=çêÇÉê= åå× K==

=527. ^=ëèì~êÉ=ã~íêáñ== [ ]áà~ ==áë==ëóããÉíêáÅ==áÑ== àááà ~~ = I==áKÉK==áí==áë=

ëóããÉíêáÅ=~Äçìí=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK===

528. ^=ëèì~êÉ=ã~íêáñ= [ ]áà~ =áë=ëâÉï-ëóããÉíêáÅ=áÑ= àááà ~~ −= K==

=

Page 121: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

111

529. aá~Öçå~ä=ã~íêáñ==áë==~=ëèì~êÉ==ã~íêáñ=ïáíÜ=~ää==ÉäÉãÉåíë==òÉêç=ÉñÅÉéí=íÜçëÉ=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äK===

530. råáí=ã~íêáñ==áë==~=Çá~Öçå~ä==ã~íêáñ==áå=ïÜáÅÜ=íÜÉ=ÉäÉãÉåíë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~ä=~êÉ=~ää=ìåáíóK=qÜÉ=ìåáí=ã~íêáñ=áë===========ÇÉåçíÉÇ=Äó=fK===

531. ^=åìää=ã~íêáñ=áë=çåÉ=ïÜçëÉ=ÉäÉãÉåíë=~êÉ=~ää=òÉêçK====

5.4 Operations with Matrices =

532. qïç=ã~íêáÅÉë=^=~åÇ=_=~êÉ=Éèì~ä=áÑI=~åÇ=çåäó=áÑI=íÜÉó=~êÉ=ÄçíÜ=çÑ==íÜÉ==ë~ãÉ==ëÜ~éÉ== åã× ==~åÇ=ÅçêêÉëéçåÇáåÖ=ÉäÉãÉåíë=~êÉ=Éèì~äK==

533. qïç=ã~íêáÅÉë==^=~åÇ=_==Å~å=ÄÉ=~ÇÇÉÇ=Eçê=ëìÄíê~ÅíÉÇF=çÑI=~åÇ=çåäó=áÑI=íÜÉó=Ü~îÉ=íÜÉ=ë~ãÉ=ëÜ~éÉ= åã× K=fÑ==

[ ]

==

ãåOãNã

åOOOON

åNNONN

áà

~~~

~~~

~~~

~^

K

MMM

K

K

I==

[ ]

==

ãåOãNã

åOOOON

åNNONN

áà

ÄÄÄ

ÄÄÄ

ÄÄÄ

Ä_

K

MMM

K

K

I==

=====

Page 122: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

112

íÜÉå==

+++

++++++

=+

ãåãåOãOãNãNã

åOåOOOOOONON

åNåNNONONNNN

Ä~Ä~Ä~

Ä~Ä~Ä~

Ä~Ä~Ä~

_^

K

MMM

K

K

K=

=534. fÑ=â=áë=~=ëÅ~ä~êI=~åÇ= [ ]áà~^ = =áë=~=ã~íêáñI=íÜÉå=

[ ]

==

ãåOãNã

åOOOON

åNNONN

áà

â~â~â~

â~â~â~

â~â~â~

â~â^

K

MMM

K

K

K=

=535. jìäíáéäáÅ~íáçå=çÑ=qïç=j~íêáÅÉë=

qïç= ã~íêáÅÉë= Å~å= ÄÉ= ãìäíáéäáÉÇ= íçÖÉíÜÉê= çåäó= ïÜÉå= íÜÉ=åìãÄÉê= çÑ= Åçäìãåë= áå= íÜÉ= Ñáêëí= áë= Éèì~ä= íç= íÜÉ= åìãÄÉê= çÑ=êçïë=áå=íÜÉ=ëÉÅçåÇK===fÑ=

[ ]

==

ãåOãNã

åOOOON

åNNONN

áà

~~~

~~~

~~~

~^

K

MMM

K

K

I==

[ ]

==

åâOåNå

âOOOON

âNNONN

áà

ÄÄÄ

ÄÄÄ

ÄÄÄ

Ä_

K

MMM

K

K

I=

=====

Page 123: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

113

íÜÉå==

==

ãâOãNã

âOOOON

âNNONN

ÅÅÄ

ÅÅÅ

ÅÅÅ

`^_

K

MMM

K

K

I==

ïÜÉêÉ==

∑=λ

λλ=+++=å

NàáåàáåàOOáàNNááà Ä~Ä~Ä~Ä~Å K =

E ãIIOINá K= X âIIOINà K= FK===qÜìë=áÑ=

[ ]

==

OPOOON

NPNONNáà ~~~

~~~~^ I= [ ]

==

P

O

N

á

Ä

Ä

Ä

Ä_ I==

íÜÉå==

=

=

POPOOONON

PNPONONNN

P

O

N

OPOOON

NPNONN

Ä~Ä~Ä~

Ä~Ä~Ä~

Ä

Ä

Ä

~~~

~~~^_ K==

=536. qê~åëéçëÉ=çÑ=~=j~íêáñ=

fÑ=íÜÉ=êçïë=~åÇ=Åçäìãåë=çÑ=~=ã~íêáñ=~êÉ=áåíÉêÅÜ~åÖÉÇI=íÜÉå=íÜÉ=åÉï=ã~íêáñ=áë=Å~ääÉÇ=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=çêáÖáå~ä=ã~íêáñK===fÑ= ^= áë= íÜÉ= çêáÖáå~ä= ã~íêáñI= áíë= íê~åëéçëÉ= áë= ÇÉåçíÉÇ= q^ = çê=

K===

537. qÜÉ=ã~íêáñ=^=áë=çêíÜçÖçå~ä=áÑ= f^^q = K===

538. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==

( ) qqq ^_^_ = K===

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CHAPTER 5. MATRICES AND DETERMINANTS

114

539. ^Çàçáåí=çÑ=j~íêáñ=fÑ=^=áë=~=ëèì~êÉ= åå× ã~íêáñI=áíë=~ÇàçáåíI=ÇÉåçíÉÇ=Äó= ^~Çà I=áë=íÜÉ=íê~åëéçëÉ=çÑ=íÜÉ=ã~íêáñ=çÑ=ÅçÑ~Åíçêë= áà` =çÑ=^W=

[ ]qáà`^~Çà = K==

=540. qê~ÅÉ=çÑ=~=j~íêáñ=

fÑ= ^= áë= ~= ëèì~êÉ= åå× ã~íêáñI= áíë= íê~ÅÉI= ÇÉåçíÉÇ= Äó= ^íê I= áë=ÇÉÑáåÉÇ=íç=ÄÉ==íÜÉ=ëìã=çÑ==íÜÉ=íÉêãë=çå=íÜÉ=äÉ~ÇáåÖ=Çá~Öçå~äW=

ååOONN ~~~^íê +++= K K==541. fåîÉêëÉ=çÑ=~=j~íêáñ=

fÑ=^=áë=~=ëèì~êÉ= åå× ã~íêáñ=ïáíÜ=~=åçåëáåÖìä~ê=ÇÉíÉêãáå~åí=^ÇÉí I=íÜÉå=áíë=áåîÉêëÉ= N^− =áë=ÖáîÉå=Äó=

^ÇÉí

^~Çà^ N =− K=

=542. fÑ=íÜÉ=ã~íêáñ=éêçÇìÅí=^_=áë=ÇÉÑáåÉÇI=íÜÉå==

( ) NNN ^_^_ −−− = K==

543. fÑ==^==áë=~=ëèì~êÉ=== åå× ==ã~íêáñI==íÜÉ==ÉáÖÉåîÉÅíçêë==u===ë~íáëÑó=íÜÉ=Éèì~íáçå=

u^u λ= I==ïÜáäÉ=íÜÉ=ÉáÖÉåî~äìÉë=λ =ë~íáëÑó=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=

Mf^ =λ− K===

===

5.5 Systems of Linear Equations ==s~êá~ÄäÉëW=ñI=óI=òI= Nñ I= KIñO =oÉ~ä=åìãÄÉêëW= KI~I~IÄI~I~I~ NONNNPON =

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CHAPTER 5. MATRICES AND DETERMINANTS

115

aÉíÉêãáå~åíëW=aI= ña I= óa I= òa ==

j~íêáÅÉëW=^I=_I=u===

544.

=+=+

OOO

NNN

ÇóÄñ~

ÇóÄñ~I==

a

añ ñ= I=

a

aó ó= =E`ê~ãÉê∞ë=êìäÉFI==

ïÜÉêÉ==

NOONOO

NN Ä~Ä~Ä~

Ä~a −== I==

NOONOO

NNñ ÄÇÄÇ

ÄÇ

ÄÇa −== I==

NOONOO

NNó Ç~Ç~

Ç~

Ç~a −== K==

=545. fÑ= Ma≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==

a

añ ñ= I=

a

aó ó= K=

fÑ= Ma= = ~åÇ= Mañ ≠ Eçê= Maó ≠ FI= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = åç==

ëçäìíáçåK=fÑ= Maaa óñ === I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= = áåÑáåáíÉäó= = ã~åó==

ëçäìíáçåëK==

546.

=++=++=++

PPPP

OOOO

NNNN

ÇòÅóÄñ~

ÇòÅóÄñ~

=ÇòÅóÄñ~

I==

a

añ ñ= I=

a

aó ó= I=

a

aò ò= =E`ê~ãÉê∞ë=êìäÉFI==

=

Page 126: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

116

ïÜÉêÉ==

PPP

OOO

NNN

ÅÄ~

ÅÄ~

ÅÄ~

a= I=

PPP

OOO

NNN

ñ

ÅÄÇ

ÅÄÇ

ÅÄÇ

a = I=

PPP

OOO

NNN

ó

ÅÇ~

ÅÇ~

ÅÇ~

a = I=

PPP

OOO

NNN

ò

ÇÄ~

ÇÄ~

ÇÄ~

a = K==

=547. fÑ= Ma≠ I=íÜÉå=íÜÉ=ëóëíÉã=Ü~ë=~=ëáåÖäÉ=ëçäìíáçåW==

a

añ ñ= I=

a

aó ó= I=

a

aò ò= K=

fÑ= Ma= =~åÇ= Mañ ≠ Eçê= Maó ≠ =çê= Maò ≠ FI=íÜÉå=íÜÉ=ëóëíÉã=

Ü~ë=åç=ëçäìíáçåK=fÑ= Maaaa òóñ ==== I= íÜÉå= íÜÉ= ëóëíÉã= Ü~ë= áåÑáåáíÉäó=

ã~åó=ëçäìíáçåëK==

548. j~íêáñ=cçêã=çÑ=~=póëíÉã=çÑ=å=iáåÉ~ê=bèì~íáçåë=áå=================å=råâåçïåë=qÜÉ=ëÉí=çÑ=äáåÉ~ê=Éèì~íáçåë==

=+++

=+++=+++

ååååOOåNNå

OååOOOONON

NååNONONNN

Äñ~ñ~ñ~

Äñ~ñ~ñ~

Äñ~ñ~ñ~

K

KKKKKKKKKKKK

K

K

=

Å~å=ÄÉ=ïêáííÉå=áå=ã~íêáñ=Ñçêã=

=

å

O

N

å

O

N

ååOåNå

åOOOON

åNNONN

Ä

Ä

Ä

ñ

ñ

ñ

~~~

~~~

~~~

MM

K

MMM

K

K

I==

áKÉK==_u^ =⋅ I==

Page 127: 1300 Math Formulas

CHAPTER 5. MATRICES AND DETERMINANTS

117

ïÜÉêÉ==

=

ååOåNå

åOOOON

åNNONN

~~~

~~~

~~~

^

K

MMM

K

K

I=

=

å

O

N

ñ

ñ

ñ

uM

I=

=

å

O

N

Ä

Ä

Ä

_M

K==

=549. pçäìíáçå=çÑ=~=pÉí=çÑ=iáåÉ~ê=bèì~íáçåë= åå× =

_^u N ⋅= − I==ïÜÉêÉ= N^− =áë=íÜÉ=áåîÉêëÉ=çÑ=^K===

Page 128: 1300 Math Formulas

118

Chapter 6

Vectors ====

sÉÅíçêëW= ìr

I= îr

I= ïr

I= êr

I=→

^_ I=£=sÉÅíçê=äÉåÖíÜW= ì

rI= îr

I=£=

råáí=îÉÅíçêëW= ár

I= àr

I= âr

=

kìää=îÉÅíçêW= Mr

=`ççêÇáå~íÉë=çÑ=îÉÅíçê= ì

rW= NNN wIvIu =

`ççêÇáå~íÉë=çÑ=îÉÅíçê= îr

W= OOO wIvIu =pÅ~ä~êëW=λ Iµ =aáêÉÅíáçå=ÅçëáåÉëW= αÅçë I= βÅçë I= γÅçë =^åÖäÉ=ÄÉíïÉÉå=íïç=îÉÅíçêëW=θ ===

6.1 Vector Coordinates =

550. råáí=sÉÅíçêë=

( )MIMINá =r

I=

( )MINIMà =r

I=

( )NIMIMâ =r

I=

Nâàá ===rrr

K=

=

551. ( ) ( ) ( )âòòàóóáññ^_ê MNMNMN

rrrr−+−+−==

=

=

Page 129: 1300 Math Formulas

CHAPTER 6. VECTORS

119

======= ==

Figure 73. =

552. ( ) ( ) ( )OMNO

MNO

MN òòóóññ^_ê −+−+−==→r

=

=

553. fÑ= ê^_r

=→

I=íÜÉå= ê_^r

−=→

K==

==

Figure 74. =

554. α= Åçëêur

I=

β= Åçëêvr

I=

γ= Åçëêwr

K=

Page 130: 1300 Math Formulas

CHAPTER 6. VECTORS

120

===== ==

Figure 75. =

555. fÑ= ( ) ( )NNNN wIvIuêwIvIuêrr

= I=íÜÉå==

Nuu = I= Nvv = I= Nww = K=====

6.2 Vector Addition =

556. îìïrrr

+= ==

== ==

Figure 76.

Page 131: 1300 Math Formulas

CHAPTER 6. VECTORS

121

== ==

Figure 77. =

557. åPON ììììïr

Krrrr

++++= =

=

== ==

Figure 78. =

558. `çããìí~íáîÉ=i~ï=ìîîìrrrr

+=+ ==

559. ^ëëçÅá~íáîÉ=i~ï=( ) ( )ïîìïîì

rrrrrr++=++ =

=560. ( )ONONON wwIvvIuuîì +++=+

rr=

======

Page 132: 1300 Math Formulas

CHAPTER 6. VECTORS

122

6.3 Vector Subtraction =

561. îìïrrr

−= =áÑ= ìïîrrr

=+ K==

==

Figure 79. =

== ==

Figure 80. =

562. ( )îìîìrrrr

−+=− ==

563. ( )MIMIMMìì ==−rrr

==

564. MM =r

=

=565. ( )ONONON wwIvvIuuîì −−−=−

rrI==

===

6.4 Scaling Vectors =

566. ìïrr

λ= =

Page 133: 1300 Math Formulas

CHAPTER 6. VECTORS

123

==

Figure 81. =

567. ìïrr

⋅λ= =

=568. ( )wIvIuì λλλ=λ

r=

=569. λ=λ ìì

rr=

=570. ( ) ììì

rrrµ+λ=µ+λ =

=571. ( ) ( ) ( )ììì

rrrλµ=λµ=µλ =

=572. ( ) îìîì

rrrrλ+λ=+λ =

===

6.5 Scalar Product =

573. pÅ~ä~ê=mêçÇìÅí=çÑ=sÉÅíçêë= ìr

=~åÇ îr

=θ⋅⋅=⋅ Åçëîìîì

rrrrI==

ïÜÉêÉ=θ =áë=íÜÉ=~åÖäÉ=ÄÉíïÉÉå=îÉÅíçêë= ìr

=~åÇ îr

K=====

Page 134: 1300 Math Formulas

CHAPTER 6. VECTORS

124

= ==

Figure 82. =

574. pÅ~ä~ê=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=fÑ= ( )NNN wIvIuì =r

I= ( )OOO wIvIuî =r

I=íÜÉå==

ONONON wwvvuuîì ++=⋅rr

K==

575. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë==fÑ= ( )NNN wIvIuì =r

I= ( )OOO wIvIuî =r

I=íÜÉå==

OO

OO

OO

ON

ON

ON

ONONON

wvuwvu

wwvvuuÅçë

++++

++=θ K=

=576. `çããìí~íáîÉ=mêçéÉêíó=

ìîîìrrrr⋅=⋅ =

=577. ^ëëçÅá~íáîÉ=mêçéÉêíó=

( ) ( ) îìîìrrrr⋅λµ=µ⋅λ =

=578. aáëíêáÄìíáîÉ=mêçéÉêíó=

( ) ïìîìïîìrrrrrrr⋅+⋅=+⋅ =

=

579. Mîì =⋅rr

=áÑ= ìr

I îr

=~êÉ=çêíÜçÖçå~ä=EO

π=θ FK=

=

580. Mîì >⋅rr

=áÑ=O

<θ< K=

=

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CHAPTER 6. VECTORS

125

581. Mîì <⋅rr

=áÑ= π<θ<πO

K=

=582. îìîì

rrrr⋅≤⋅ =

=583. îìîì

rrrr⋅=⋅ =áÑ= ì

rI îr

=~êÉ=é~ê~ääÉä=E M=θ FK=

=584. fÑ= ( )NNN wIvIuì =

rI=íÜÉå==

ON

ON

ON

OO wvuìììì ++===⋅rrrr

K=

=

585. Nââààáá =⋅=⋅=⋅rrrrrr

==

586. Máââààá =⋅=⋅=⋅rrrrrr

====

6.6 Vector Product =

587. sÉÅíçê=mêçÇìÅí=çÑ=sÉÅíçêë= ìr

=~åÇ îr

=ïîìrrr

=× I=ïÜÉêÉ==

• θ⋅⋅= ëáåîìïrrr

I=ïÜÉêÉ=O

≤θ≤ X=

• ìïrr

⊥ = ~åÇ= îïrr

⊥ X=• =sÉÅíçêë= ì

rI= îr

I= ïr

=Ñçêã=~=êáÖÜí-Ü~åÇÉÇ=ëÅêÉïK==

Page 136: 1300 Math Formulas

CHAPTER 6. VECTORS

126

======= ==

Figure 83. =

588. OOO

NNN

wvu

wvu

âàá

îìï

rrr

rrr=×= =

=

589.

−=×=

OO

NN

OO

NN

OO

NN

vu

vuI

wu

wuI

wv

wvîìïrrr

=

=590. θ⋅⋅=×= ëáåîìîìp

rrrr=EcáÖKUPF=

=591. ^åÖäÉ=_ÉíïÉÉå=qïç=sÉÅíçêë=EcáÖKUPF=

îì

îìëáå rr

rr

⋅×

=θ =

=592. kçåÅçããìí~íáîÉ=mêçéÉêíó=

( )ìîîìrrrr

×−=× ===

593. ^ëëçÅá~íáîÉ=mêçéÉêíó=( ) ( ) îìîì

rrrr×λµ=µ×λ =

==

Page 137: 1300 Math Formulas

CHAPTER 6. VECTORS

127

594. aáëíêáÄìíáîÉ=mêçéÉêíó=( ) ïìîìïîì

rrrrrrr×+×=+× =

=595. Mîì

rrr=× =áÑ= ì

r=~åÇ= î

r=~êÉ=é~ê~ääÉä=E M=θ FK=

=

596. Mââààáárrrrrrr

=×=×=× ==

597. âàárrr

=× I= áâàrrr

=× I= àáârrr

=× ====

6.7 Triple Product =

598. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=[ ] ( ) ( ) ( )îìïìïîïîìïîì

rrrrrrrrrrrr×⋅=×⋅=×⋅= =

=599. [ ] [ ] [ ] [ ] [ ] [ ]îïììîïïìîìïîîìïïîì

rrrrrrrrrrrrrrrrrr−=−=−=== =

=600. ( ) [ ]ïîìâïîìâ

rrrrrr=×⋅ =

=601. pÅ~ä~ê=qêáéäÉ=mêçÇìÅí=áå=`ççêÇáå~íÉ=cçêã=

( )PPP

OOO

NNN

wvu

wvu

wvu

ïîì =×⋅rrr

I==

ïÜÉêÉ==( )NNN wIvIuì =

rI= ( )OOO wIvIuî =r

I= ( )PPP wIvIuï =r

K==

=602. sçäìãÉ=çÑ=m~ê~ääÉäÉéáéÉÇ=

( )ïîìsrrr

×⋅= =

=

Page 138: 1300 Math Formulas

CHAPTER 6. VECTORS

128

============ ==

Figure 84. =

603. sçäìãÉ=çÑ=móê~ãáÇ=

( )ïîìS

Ns

rrr×⋅= =

=

==

Figure 85. =

604. fÑ== ( ) Mïîì =×⋅rrr

I=íÜÉå=íÜÉ=îÉÅíçêë== ìr

I= îr

I=~åÇ= ïr

=~êÉ=äáåÉ~êäó=ÇÉéÉåÇÉåí=I=ëç= îìï

rrrµ+λ= =Ñçê=ëçãÉ=ëÅ~ä~êë=λ =~åÇ=µ K==

=605. fÑ== ( ) Mïîì ≠×⋅

rrrI=íÜÉå=íÜÉ=îÉÅíçêë== ì

rI= îr

I=~åÇ= ïr

=~êÉ=äáåÉ~êäó=áåÇÉéÉåÇÉåíK==

Page 139: 1300 Math Formulas

CHAPTER 6. VECTORS

129

606. sÉÅíçê=qêáéäÉ=mêçÇìÅí=( ) ( ) ( )ïîìîïìïîì

rrrrrrrrr⋅−⋅=×× ==

========

Page 140: 1300 Math Formulas

130

Chapter 7

Analytic Geometry ====

7.1 One-Dimensional Coordinate System =mçáåí=ÅççêÇáå~íÉëW= Mñ I= Nñ I= Oñ I= Mó I= Nó I= Oó =

oÉ~ä=åìãÄÉêW=λ ==aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç===

607. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=

ONNO ññññ^_Ç −=−== =

=

==

Figure 86. =

608. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =

λ+λ+

=N

ñññ ON

M I=`_

^`=λ I= N−≠λ K=

=

======== ==

Figure 87.

Page 141: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

131

609. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=

O

ñññ ON

M

+= I= N=λ K=

===

7.2 Two-Dimensional Coordinate System =mçáåí=ÅççêÇáå~íÉëW= Mñ I= Nñ I= Oñ I= Mó I= Nó I= Oó =

mçä~ê=ÅççêÇáå~íÉëW= ϕIê =oÉ~ä=åìãÄÉêW=λ ==mçëáíáîÉ=êÉ~ä=åìãÄÉêëW=~I=ÄI=ÅI==aáëí~åÅÉ=ÄÉíïÉÉå=íïç=éçáåíëW=Ç=^êÉ~W=p===

610. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=

( ) ( )ONOO

NO óóññ^_Ç −+−== =

=

==

Figure 88.

Page 142: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

132

611. aáîáÇáåÖ=~=iáåÉ=pÉÖãÉåí=áå=íÜÉ=o~íáç=λ =

λ+λ+

=N

ñññ ON

M I=λ+λ+

=N

óóó ON

M I==

`_

^`=λ I= N−≠λ K=

=

======= ==

Figure 89. ==

Page 143: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

133

======= ==

Figure 90. =

612. jáÇéçáåí=çÑ=~=iáåÉ=pÉÖãÉåí=

O

ñññ ON

M

+= I=

O

óóó ON

M

+= I= N=λ K=

=613. `ÉåíêçáÇ=EfåíÉêëÉÅíáçå=çÑ=jÉÇá~åëF=çÑ=~=qêá~åÖäÉ=

P

ññññ PON

M

++= I=

P

óóóó PON

M

++= I==

ïÜÉêÉ== ( )NN óIñ^ I== ( )OO óIñ_ I==~åÇ== ( )PP óIñ` ==~êÉ=îÉêíáÅÉë=çÑ=

íÜÉ=íêá~åÖäÉ= ^_` K= ==

Page 144: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

134

========= ==

Figure 91. =

614. fåÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^åÖäÉ=_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=

ÅÄ~

ÅñÄñ~ññ PON

M ++++

= I=ÅÄ~

ÅóÄó~óó PON

M ++++

= I==

ïÜÉêÉ= _`~ = I= `^Ä = I= ^_Å = K===

======== ==

Figure 92.

Page 145: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

135

615. `áêÅìãÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=íÜÉ=páÇÉ=mÉêéÉåÇáÅìä~ê======================_áëÉÅíçêëF=çÑ=~=qêá~åÖäÉ=

Nóñ

Nóñ

Nóñ

O

Nóóñ

Nóóñ

Nóóñ

ñ

PP

OO

NN

POP

OP

OOO

OO

NON

ON

M

+++

= I=

Nóñ

Nóñ

Nóñ

O

Nóññ

Nóññ

Nóññ

ó

PP

OO

NN

OP

OPP

OO

OOO

ON

ONN

M

+++

= =

=

======== ===

Figure 93. =======

Page 146: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

136

616. lêíÜçÅÉåíÉê=EfåíÉêëÉÅíáçå=çÑ=^äíáíìÇÉëF=çÑ=~=qêá~åÖäÉ=

Nóñ

Nóñ

Nóñ

Nóññó

Nóññó

Nóññó

ñ

PP

OO

NN

OPONP

OONPO

ONPON

M

+++

= I=

Nóñ

Nóñ

Nóñ

Nñóóñ

Nñóóñ

Nñóóñ

ó

PP

OO

NN

PONOP

ONPOO

NPOON

M

+++

= =

=

====== ==

Figure 94. =

617. ^êÉ~=çÑ=~=qêá~åÖäÉ=

( ) ( )NPNP

NONO

PP

OO

NN

óóññ

óóññ

O

N

Nóñ

Nóñ

Nóñ

O

Np

−−−−

±=±= =

===

Page 147: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

137

618. ^êÉ~=çÑ=~=nì~Çêáä~íÉê~ä=

( ) ( )( ) ( )( )[ ++−++−±= POPOONON óóññóóññO

Np =

( )( ) ( )( )]NQNQQPQP óóññóóññ +−++−+ =

=

=== ==

Figure 95. =kçíÉW=få=Ñçêãìä~ë=SNTI=SNU=ïÉ=ÅÜççëÉ=íÜÉ=ëáÖå=EHF=çê=E¥F=ëç=íÜ~í=íç=ÖÉí=~=éçëáíáîÉ=~åëïÉê=Ñçê=~êÉ~K==

=619. aáëí~åÅÉ=_ÉíïÉÉå=qïç=mçáåíë=áå=mçä~ê=`ççêÇáå~íÉë=

( )NOONOO

ON ÅçëêêOêê^_Ç ϕ−ϕ−+== =

=

Page 148: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

138

==

Figure 96. =

620. `çåîÉêíáåÖ=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=íç=mçä~ê=`ççêÇáå~íÉë=ϕ= Åçëêñ I= ϕ= ëáåêó K=

=

==

Figure 97. =

621. `çåîÉêíáåÖ=mçä~ê=`ççêÇáå~íÉë=íç=oÉÅí~åÖìä~ê=`ççêÇáå~íÉë=

OO óñê += I=ñ

óí~å =ϕ K=

Page 149: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

139

7.3 Straight Line in Plane =mçáåí=ÅççêÇáå~íÉëW=uI=vI=ñI= Mñ I= Nñ I== Mó I= Nó I= N~ I= O~ I=£==

oÉ~ä=åìãÄÉêëW=âI=~I=ÄI=éI=íI=^I=_I=`I= N^ I= O^ I=£=^åÖäÉëW=α I=β =^åÖäÉ=ÄÉíïÉÉå=íïç=äáåÉëW=ϕ =kçêã~ä=îÉÅíçêW= å

r=

mçëáíáçå=îÉÅíçêëW= êr

I= ~r

I= Är

===

622. dÉåÉê~ä=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=M`_ó^ñ =++ =

=623. kçêã~ä=sÉÅíçê=íç=~=píê~áÖÜí=iáåÉ=

qÜÉ=îÉÅíçê= ( )_I^år

=áë=åçêã~ä=íç=íÜÉ=äáåÉ= M`_ó^ñ =++ K==

==

Figure 98. =

624. bñéäáÅáí=bèì~íáçå=çÑ=~=píê~áÖÜí=iáåÉ=EpäçéÉ-fåíÉêÅÉéí=cçêãF=Äâñó += K==

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CHAPTER 7. ANALYTIC GEOMETRY

140

qÜÉ=Öê~ÇáÉåí=çÑ=íÜÉ=äáåÉ=áë= α= í~åâ K==

==

Figure 99. =

625. dê~ÇáÉåí=çÑ=~=iáåÉ==

NO

NO

ññ

óóí~åâ

−−

=α= =

=

==

Figure 100.

Page 151: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

141

626. bèì~íáçå=çÑ=~=iáåÉ=dáîÉå=~=mçáåí=~åÇ=íÜÉ=dê~ÇáÉåí=( )MM ññâóó −+= I==

ïÜÉêÉ=â=áë=íÜÉ=Öê~ÇáÉåíI= ( )MM óIñm =áë=~=éçáåí=çå=íÜÉ=äáåÉK=

=

==

Figure 101. =

627. bèì~íáçå=çÑ=~=iáåÉ=qÜ~í=m~ëëÉë=qÜêçìÖÜ=qïç=mçáåíë=

NO

N

NO

N

ññ

ññ

óó

óó

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CHAPTER 7. ANALYTIC GEOMETRY

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CHAPTER 7. ANALYTIC GEOMETRY

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Page 190: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

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181

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Page 192: 1300 Math Formulas

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Page 194: 1300 Math Formulas

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Page 195: 1300 Math Formulas

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Page 196: 1300 Math Formulas

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Page 197: 1300 Math Formulas

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Page 198: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

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Page 199: 1300 Math Formulas

CHAPTER 7. ANALYTIC GEOMETRY

189

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CHAPTER 7. ANALYTIC GEOMETRY

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Page 201: 1300 Math Formulas

191

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CHAPTER 8. DIFFERENTIAL CALCULUS

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CHAPTER 8. DIFFERENTIAL CALCULUS

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Page 204: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

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CHAPTER 8. DIFFERENTIAL CALCULUS

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Page 206: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

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Page 208: 1300 Math Formulas

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Page 209: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

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Page 211: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

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Page 212: 1300 Math Formulas

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=Figure 167.

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203

742. eóéÉêÄçäáÅ=`çëÉÅ~åí=cìåÅíáçå==

ñÅëÅÜó = I=ññ ÉÉ

O

ñëáåÜ

NñÅëÅÜó −−

=== I= oñ∈ I= Mñ ≠ K=

=

====== ==

Figure 168. =

743. fåîÉêëÉ=eóéÉêÄçäáÅ=páåÉ=cìåÅíáçå==ñ~êÅëáåÜó = I= oñ∈ K=

=

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204

===== ==

Figure 169. =

744. fåîÉêëÉ=eóéÉêÄçäáÅ=`çëáåÉ=cìåÅíáçå==ñ~êÅÅçëÜó = I= [ )∞∈ INñ K=

=

===== ==

Figure 170. =

745. fåîÉêëÉ=eóéÉêÄçäáÅ=q~åÖÉåí=cìåÅíáçå==ñ~êÅí~åÜó = I= ( )NINñ −∈ K=

=

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205

===== ==

Figure 171. =

746. fåîÉêëÉ=eóéÉêÄçäáÅ=`çí~åÖÉåí=cìåÅíáçå==ñ~êÅÅçíÜó = I= ( ) ( )∞∪−∞−∈ INNIñ K==

=

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206

===== ==

Figure 172. =

747. fåîÉêëÉ=eóéÉêÄçäáÅ=pÉÅ~åí=cìåÅíáçå==ñ~êÅëÉÅÜó = I= ( ]NIMñ∈ K=

=

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207

=Figure 173.

=748. fåîÉêëÉ=eóéÉêÄçäáÅ=`çëÉÅ~åí=cìåÅíáçå==

ñ~êÅÅëÅÜó = I= oñ∈ I= Mñ ≠ K==

= Figure 174.

=

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208

8.2 Limits of Functions =cìåÅíáçåëW= ( )ñÑ I= ( )ñÖ =^êÖìãÉåíW=ñ=oÉ~ä=Åçåëí~åíëW=~I=â===

749. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã~ñ~ñ~ñ →→→

+=+ =

=750. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã

~ñ~ñ~ñ →→→−=− =

=751. ( ) ( )[ ] ( ) ( )ñÖäáãñÑäáãñÖñÑäáã

~ñ~ñ~ñ →→→⋅=⋅ =

=

752. ( )( )

( )( )ñÖäáã

ñÑäáã

ñÖ

ñÑäáã

~ñ→

→= I=áÑ= ( ) MñÖäáã

~ñ≠

→K=

=753. ( )[ ] ( )ñÑäáãâñâÑäáã

~ñ~ñ →→= =

=754. ( )( ) ( )( )ñÖäáãÑñÖÑäáã

~ñ~ñ →→= =

=755. ( ) ( )~ÑñÑäáã

~ñ=

→I=áÑ=íÜÉ=ÑìåÅíáçå= ( )ñÑ =áë=Åçåíáåìçìë=~í= ~ñ = K=

=

756. Nñ

ñëáåäáã

Mñ=

→=

=

757. Nñ

ñí~åäáã

Mñ=

→=

=

758. Nñ

ñëáåäáã

N

Mñ=

→=

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209

759. Nñ

ñí~åäáã

N

Mñ=

→=

=

760. ( )N

ñ

ñNäåäáã

Mñ=

+→

=

=

761. Éñ

NNäáã

ñ

ñ=

+

∞→=

=

762. âñ

ñÉ

ñ

âNäáã =

+

∞→=

=763. N~äáã ñ

Mñ=

→=

===

8.3 Definition and Properties of the Derivative =cìåÅíáçåëW=ÑI=ÖI=óI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=oÉ~ä=Åçåëí~åíW=â=^åÖäÉW=α ===

764. ( ) ( ) ( )Çñ

Çó

ñ

óäáã

ñ

ñÑññÑäáãñó

MñMñ=

∆∆

=∆

−∆+=′

→∆→∆==

=

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210

== ==

Figure 175. =

765. α= í~åÇñ

Çó==

=

766. ( )Çñ

Çî

Çñ

Çì

Çñ

îìÇ+=

+=

=

767. ( )Çñ

Çî

Çñ

Çì

Çñ

îìÇ−=

−=

=

768. ( )Çñ

Çìâ

Çñ

âìÇ= =

=769. mêçÇìÅí=oìäÉ=

( )Çñ

Çîìî

Çñ

Çì

Çñ

îìÇ⋅+⋅=

⋅==

==

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211

770. nìçíáÉåí=oìäÉ=

OîÇñ

Çîìî

Çñ

Çì

î

ì

Çñ

Ç ⋅−⋅=

=

=771. `Ü~áå=oìäÉ=

( )( )ñÖÑó = I= ( )ñÖì = I==

Çñ

Çì

Çì

Çó

Çñ

Çó⋅= K=

=772. aÉêáî~íáîÉ=çÑ=fåîÉêëÉ=cìåÅíáçå=

Çó

ÇñN

Çñ

Çó= I==

ïÜÉêÉ= ( )óñ áë=íÜÉ=áåîÉêëÉ=ÑìåÅíáçå=çÑ= ( )ñó K===

773. oÉÅáéêçÅ~ä=oìäÉ=

OóÇñ

Çó

ó

N

Çñ

Ç−=

=

=774. içÖ~êáíÜãáÅ=aáÑÑÉêÉåíá~íáçå=

( )ñÑó = I= ( )ñÑäåóäå = I==

( ) ( )[ ]ñÑäåÇñ

ÇñÑ

Çñ

Çó⋅= K=

==

8.4 Table of Derivatives =fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=oÉ~ä=Åçåëí~åíëW=`I=~I=ÄI=Å=k~íìê~ä=åìãÄÉêW=å=

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212

775. ( ) M`Çñ

Ç= =

=

776. ( ) NñÇñ

Ç= =

=

777. ( ) ~Ä~ñÇñ

Ç=+ =

=

778. ( ) Ä~ñÅÄñ~ñÇñ

Ç O +=++ =

=

779. ( ) Nåå åññÇñ

Ç −= =

=

780. ( )Nå

å

ñ

åñ

Çñ

Ç+

− −= =

=

781. Oñ

N

ñ

N

Çñ

Ç−=

=

=

782. ( )ñO

Çñ

Ç= =

=

783. ( )å Nå

å

ñå

Çñ

Ç−

= =

=

784. ( )ñ

Nñäå

Çñ

Ç= =

=

785. ( )~äåñ

NñäçÖ

Çñ

Ç~ = I= M~ > I= N~ ≠ K=

=

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213

786. ( ) ~äå~~Çñ

Ç ññ = I= M~ > I= N~ ≠ K=

=

787. ( ) ññ ÉÉÇñ

Ç= =

=

788. ( ) ñÅçëñëáåÇñ

Ç= =

=

789. ( ) ñëáåñÅçëÇñ

Ç−= =

=

790. ( ) ñëÉÅñÅçë

Nñí~å

Çñ

Ç O

O== =

=

791. ( ) ñÅëÅñëáå

NñÅçí

Çñ

Ç O

O−=−= =

=

792. ( ) ñëÉÅñí~åñëÉÅÇñ

Ç⋅= =

=

793. ( ) ñÅëÅñÅçíñÅëÅÇñ

Ç⋅−= =

=

794. ( )OñN

Nñ~êÅëáå

Çñ

Ç

−= =

=

795. ( )OñN

Nñ~êÅÅçë

Çñ

Ç

−−= =

=

796. ( )OñN

Nñ~êÅí~å

Çñ

Ç

+= =

=

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214

797. ( )OñN

NñÅçí~êÅ

Çñ

Ç

+−= =

=

798. ( )Nññ

NñëÉÅ~êÅ

Çñ

ÇO −

= =

=

799. ( )Nññ

NñÅëÅ~êÅ

Çñ

ÇO −

−= =

=

800. ( ) ñÅçëÜñëáåÜÇñ

Ç= =

=

801. ( ) ñëáåÜñÅçëÜÇñ

Ç= =

=

802. ( ) ñëÉÅÜñÅçëÜ

Nñí~åÜ

Çñ

Ç O

O== =

=

803. ( ) ñÅëÅÜñëáåÜ

NñÅçíÜ

Çñ

Ç O

O−=−= =

=

804. ( ) ñí~åÜñëÉÅÜñëÉÅÜÇñ

Ç⋅−= =

=

805. ( ) ñÅçíÜñÅëÅÜñÅëÅÜÇñ

Ç⋅−= =

=

806. ( )Nñ

Nñ=~êÅëáåÜ

Çñ

ÇO +

= =

=

807. ( )Nñ

Nñ=~êÅÅçëÜ

Çñ

ÇO −

= =

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215

808. ( )OñN

Nñ=~êÅí~åÜ

Çñ

Ç

−= I= Nñ < K=

=

809. ( )Nñ

Nñ=~êÅÅçíÜ

Çñ

ÇO −

−= I= Nñ > K=

=

810. ( )Çñ

Çîìäåì

Çñ

Çìîìì

Çñ

Ç îNîî ⋅+⋅= − =

===

8.5 Higher Order Derivatives =cìåÅíáçåëW=ÑI=óI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=k~íìê~ä=åìãÄÉêW=å===

811. pÉÅçåÇ=ÇÉêáî~íáîÉ=

( )O

O

Çñ

óÇ

Çñ

Çó

Çñ

Ç

Çñ

ÇóÑÑ =

=′

=′′=′′ =

=812. eáÖÜÉê-lêÇÉê=ÇÉêáî~íáîÉ=

( ) ( ) ( )( ) ′=== −Nåå

å

åå Ñó

Çñ

óÇÑ =

=

813. ( )( ) ( ) ( )ååå îìîì +=+ ==

814. ( )( ) ( ) ( )ååå îìîì −=− ==

815. iÉáÄåáíò∞ë=cçêãìä~ë=

( ) îìîìOîììî ′′+′′+′′=′′ =

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216

( ) îìîìPîìPîììî ′′′+′′′+′′′+′′′=′′′ =

( )( ) ( ) ( ) ( ) ( ) ( )åOåNååå ìîîìON

Nååîåìîììî ++′′

⋅−

+′+= −− K =

=

816. ( )( )

( )åãåã ñ

>åã

>ãñ −

−= =

=

817. ( )( )>åñ

åå = ==

818. ( )( ) ( ) ( )~äåñ

>NåNñäçÖ

å

Nåå

~

−−=

=

=

819. ( )( ) ( ) ( )å

Nåå

ñ

>NåNñäå

−−=

=

=

820. ( )( )~äå~~ åñåñ = =

=

821. ( )( ) ñåñ ÉÉ = ==

822. ( )( )~äå~ã~ åãñååãñ = =

=

823. ( )( )

π

+=O

åñëáåñëáå å =

=

824. ( )( )

π

+=O

åñÅçëñÅçë å =

===

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217

8.6 Applications of Derivative =cìåÅíáçåëW=ÑI=ÖI=ó=mçëáíáçå=çÑ=~å=çÄàÉÅíW=ë==sÉäçÅáíóW=î=^ÅÅÉäÉê~íáçåW=ï=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=qáãÉW=í=k~íìê~ä=åìãÄÉêW=å===

825. sÉäçÅáíó=~åÇ=^ÅÅÉäÉê~íáçå=( )íÑë = =áë=íÜÉ=éçëáíáçå=çÑ=~å=çÄàÉÅí=êÉä~íáîÉ=íç=~=ÑáñÉÇ=

ÅççêÇáå~íÉ=ëóëíÉã=~í=~=íáãÉ=íI==( )íÑëî ′=′= =áë=íÜÉ=áåëí~åí~åÉçìë=îÉäçÅáíó=çÑ=íÜÉ=çÄàÉÅíI=

( )íÑëîï ′′=′′=′= =áë=íÜÉ=áåëí~åí~åÉçìë=~ÅÅÉäÉê~íáçå=çÑ=íÜÉ=çÄàÉÅíK===

826. q~åÖÉåí=iáåÉ=( )( )MMM ñññÑóó −′=− =

=

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218

==

Figure 176. =

827. kçêã~ä=iáåÉ=

( )( )M

M

M ñññÑ

Nóó −

′−=− =EcáÖ=NTSF=

=828. fåÅêÉ~ëáåÖ=~åÇ=aÉÅêÉ~ëáåÖ=cìåÅíáçåëK==

fÑ= ( ) MñÑ M >′ I=íÜÉå=ÑEñF=áë=áåÅêÉ~ëáåÖ=~í= Mñ K=EcáÖ=NTTI= Nññ < I=

ññO < FI=fÑ= ( ) MñÑ M <′ I=íÜÉå=ÑEñF=áë=ÇÉÅêÉ~ëáåÖ=~í= Mñ K=EcáÖ=NTTI=

ON ñññ << FI=fÑ= ( )MñÑ ′ =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK==

=

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219

==

Figure 177. =

829. içÅ~ä=ÉñíêÉã~=^=ÑìåÅíáçå=ÑEñF=Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í= Nñ =áÑ=~åÇ=çåäó=áÑ=íÜÉêÉ=Éñáëíë=ëçãÉ=áåíÉêî~ä=Åçåí~áåáåÖ= Nñ =ëìÅÜ=íÜ~í=( ) ( )ñÑñÑ N ≥ =Ñçê=~ää=ñ=áå=íÜÉ=áåíÉêî~ä=EcáÖKNTTFK==

=^=ÑìåÅíáçå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ =áÑ=~åÇ=çåäó=áÑ=íÜÉêÉ=Éñáëíë=ëçãÉ=áåíÉêî~ä=Åçåí~áåáåÖ= Oñ =ëìÅÜ=íÜ~í=( ) ( )ñÑñÑ O ≤ =Ñçê=~ää=ñ=áå=íÜÉ=áåíÉêî~ä=EcáÖKNTTFK=

=830. `êáíáÅ~ä=mçáåíë=

^=ÅêáíáÅ~ä=éçáåí=çå=ÑEñF=çÅÅìêë=~í= Mñ =áÑ=~åÇ=çåäó=áÑ=ÉáíÜÉê=

( )MñÑ ′ =áë=òÉêç=çê=íÜÉ=ÇÉêáî~íáîÉ=ÇçÉëå∞í=ÉñáëíK=

=831. cáêëí=aÉêáî~íáîÉ=qÉëí=Ñçê=içÅ~ä=bñíêÉã~K=

fÑ=ÑEñF=áë==áåÅêÉ~ëáåÖ==E ( ) MñÑ >′ F=Ñçê==~ää==ñ==áå==ëçãÉ==áåíÉêî~ä=( ]NñI~ ==~åÇ==ÑEñF==áë==ÇÉÅêÉ~ëáåÖ==E ( ) MñÑ <′ F==Ñçê=~ää==ñ=áå=ëçãÉ=áåíÉêî~ä= = [ )ÄIñN I= = íÜÉå= ÑEñF= Ü~ë= ~= = äçÅ~ä= ã~ñáãìã= = ~í= = Nñ =EcáÖKNTTFK==

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220

832. fÑ=ÑEñF=áë=ÇÉÅêÉ~ëáåÖ=E ( ) MñÑ <′ F=Ñçê=~ää=ñ=áå=ëçãÉ=áåíÉêî~ä=( ]OñI~ =~åÇ=ÑEñF=áë=áåÅêÉ~ëáåÖ=E ( ) MñÑ >′ F=Ñçê=~ää=ñ=áå=ëçãÉ=áåíÉêî~ä= [ )ÄIñ O I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ K==EcáÖKNTTFK==

833. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=içÅ~ä=bñíêÉã~K=fÑ= ( ) MñÑ N =′ =~åÇ= ( ) MñÑ N <′′ I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í== Nñ K=fÑ= ( ) MñÑ O =′ =~åÇ= ( ) MñÑ O >′′ I=íÜÉå=ÑEñF=Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= Oñ K=EcáÖKNTTF==

834. `çåÅ~îáíóK==ÑEñF= áë= = ÅçåÅ~îÉ= ìéï~êÇ= ~í= = Mñ = = áÑ= = ~åÇ= = çåäó= = áÑ= = ( )ñÑ ′ = áë============

áåÅêÉ~ëáåÖ=~í= Mñ =EcáÖKNTTI= ññ P < FK===

ÑEñF= áë= =ÅçåÅ~îÉ= =Ççïåï~êÇ=~í= = Mñ = = áÑ=~åÇ=çåäó= áÑ= = ( )ñÑ ′ = = áë===============

ÇÉÅêÉ~ëáåÖ=~í= Mñ K=EcáÖKNTTI= Pññ < FK===

=835. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=`çåÅ~îáíóK==

fÑ= ( ) MñÑ M >′′ I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=ìéï~êÇ=~í= Mñ K==

fÑ= ( ) MñÑ M <′′ I=íÜÉå=ÑEñF=áë=ÅçåÅ~îÉ=Ççïåï~êÇ=~í= Mñ K=

fÑ= ( )ñÑ ′′ =ÇçÉë=åçí=Éñáëí=çê=áë=òÉêçI=íÜÉå=íÜÉ=íÉëí=Ñ~áäëK==

836. fåÑäÉÅíáçå=mçáåíë=fÑ= = ( )PñÑ ′ = =Éñáëíë==~åÇ== ( )ñÑ ′′ = =ÅÜ~åÖÉë=ëáÖå=~í= Pññ = I= =íÜÉå=

íÜÉ= éçáåí= ( )( )PP ñÑIñ = áë= ~å= áåÑäÉÅíáçå= éçáåí= çÑ= íÜÉ= Öê~éÜ= çÑ=

( )ñÑ K=fÑ= ( )PñÑ ′′ =Éñáëíë=~í=íÜÉ=áåÑäÉÅíáçå=éçáåíI=íÜÉå= ( ) MñÑ P =′′ =

EcáÖKNTTFK==

837. i∞eçéáí~ä∞ë=oìäÉ=

( )( )

( )( )ñÖ

ñÑäáã

ñÖ

ñÑäáã

ÅñÅñ ′′

=→→

=áÑ= ( ) ( )∞

==→→

MñÖäáãñÑäáã

ÅñÅñK==

=

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221

8.7 Differential =cìåÅíáçåëW=ÑI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉW=ñ=aÉêáî~íáîÉ=çÑ=~=ÑìåÅíáçåW= ( )ñó′ I= ( )ñÑ ′ =oÉ~ä=Åçåëí~åíW=`=aáÑÑÉêÉåíá~ä=çÑ=ÑìåÅíáçå= ( )ñÑó = W=Çó=aáÑÑÉêÉåíá~ä=çÑ=ñW=Çñ=pã~ää=ÅÜ~åÖÉ=áå=ñW= ñ∆ =pã~ää=ÅÜ~åÖÉ=áå=óW= ó∆ ===

838. ÇñóÇó ′= ==

839. ( ) ( ) ( ) ññÑñÑññÑ ∆′+=∆+ ==

==

Figure 178.

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222

840. pã~ää=`Ü~åÖÉ=áå=ó=( ) ( )ñÑññÑó −∆+=∆ =

=841. ( ) ÇîÇìîìÇ +=+ =

=842. ( ) ÇîÇìîìÇ −=− =

=843. ( ) `Çì`ìÇ = =

=844. ( ) ìÇîîÇììîÇ += =

=

845. Oî

ìÇîîÇì

î

ìÇ

−=

=

===

8.8 Multivariable Functions =cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= ( )óIñò I= ( )óIñÑ I= ( )óIñÖ I= ( )óIñÜ ==^êÖìãÉåíëW=ñI=óI=í=pã~ää=ÅÜ~åÖÉë=áå=ñI=óI=òI=êÉëéÉÅíáîÉäóW= ñ∆ I= ó∆ I= ò∆ K===

846. cáêëí=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ñ=

ñÑñ

Ñ=

∂∂

=E~äëç= ñòñ

ò=

∂∂

FI=

qÜÉ=é~êíá~ä=ÇÉêáî~íáîÉ=ïáíÜ=êÉëéÉÅí=íç=ó=

óÑó

Ñ=

∂∂

=E~äëç= óòó

ò=

∂∂

FK=

==

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223

847. pÉÅçåÇ=lêÇÉê=m~êíá~ä=aÉêáî~íáîÉë=

ññO

O

Ññ

Ñ

ñ

Ñ

ñ=

∂∂

=

∂∂

∂∂

I==

óóO

O

Ñó

Ñ

ó

Ñ

ó=

∂∂

=

∂∂

∂∂

I==

ñó

O

Ññó

Ñ

ñ

Ñ

ó=

∂∂∂

=

∂∂

∂∂

I==

óñ

O

Ñóñ

Ñ

ó

Ñ

ñ=

∂∂∂

=

∂∂

∂∂

K==

fÑ=íÜÉ=ÇÉêáî~íáîÉë=~êÉ=ÅçåíáåìçìëI=íÜÉå==

óñ

Ñ

ñó

Ñ OO

∂∂∂

=∂∂

∂K==

=848. `Ü~áå=oìäÉë==

fÑ= ( ) ( )( )óIñÜÖóIñÑ = =EÖ=áë=~=ÑìåÅíáçå=çÑ=çåÉ=î~êá~ÄäÉ=ÜFI=íÜÉå==

( )( )ñ

ÜóIñÜÖ

ñ

Ñ

∂∂′=

∂∂

I= ( )( )ó

ÜóIñÜÖ

ó

Ñ

∂∂′=

∂∂

K==

=

fÑ= ( ) ( ) ( )( )íóIíñÑíÜ = I=íÜÉå= ( )Çí

Çó

ó

Ñ

Çí

Çñ

ñ

ÑíÜ

∂∂

+∂∂

=′ K==

=fÑ= ( ) ( )( )îIìóIîIìñÑò = I=íÜÉå==

ì

ó

ó

Ñ

ì

ñ

ñ

Ñ

ì

ò

∂∂

∂∂

+∂∂

∂∂

=∂∂

I=î

ó

ó

Ñ

î

ñ

ñ

Ñ

î

ò

∂∂

∂∂

+∂∂

∂∂

=∂∂

K==

=849. pã~ää=`Ü~åÖÉë=

óó

Ññ

ñ

Ñò ∆

∂∂

+∆∂∂

≈∆ =

==

Page 234: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

224

850. içÅ~ä=j~ñáã~=~åÇ=jáåáã~=( )óIñÑ =Ü~ë=~=äçÅ~ä=ã~ñáãìã=~í= ( )MM óIñ =áÑ= ( ) ( )MM óIñÑóIñÑ ≤ =

Ñçê=~ää= ( )óIñ =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= ( )MM óIñ K==

=( )óIñÑ =Ü~ë=~=äçÅ~ä=ãáåáãìã=~í= ( )MM óIñ =áÑ= ( ) ( )MM óIñÑóIñÑ ≥ =

Ñçê=~ää= ( )óIñ =ëìÑÑáÅáÉåíäó=ÅäçëÉ=íç= ( )MM óIñ K=

=851. pí~íáçå~êó=mçáåíë=

Ñ

ñ

Ñ=

∂∂

=∂∂

K=

içÅ~ä=ã~ñáã~=~åÇ=äçÅ~ä=ãáåáã~=çÅÅìê=~í=ëí~íáçå~êó=éçáåíëK===

852. p~ÇÇäÉ=mçáåí=^=ëí~íáçå~êó= =éçáåí= =ïÜáÅÜ= = áë= =åÉáíÜÉê= =~= = äçÅ~ä= =ã~ñáãìã=åçê=~=äçÅ~ä=ãáåáãìã==

853. pÉÅçåÇ=aÉêáî~íáîÉ=qÉëí=Ñçê=pí~íáçå~êó=mçáåíë=

iÉí= ( )MM óIñ =ÄÉ=~=ëí~íáçå~êó=éçáåí=E Mó

Ñ

ñ

Ñ=

∂∂

=∂∂

FK==

( ) ( )( ) ( )MMóóMMóñ

MMñóMMññ

óIñÑóIñÑ

óIñÑóIñÑa = K==

=fÑ= Ma> I= ( ) MóIñÑ MMññ > I== ( )MM óIñ ==áë=~=éçáåí=çÑ=äçÅ~ä=ãáåáã~K=

fÑ= Ma> I= ( ) MóIñÑ MMññ < I== ( )MM óIñ ==áë=~=éçáåí=çÑ=äçÅ~ä=ã~ñáã~K=

fÑ= Ma< I= ( )MM óIñ =áë=~=ë~ÇÇäÉ=éçáåíK=

fÑ= Ma= I=íÜÉ=íÉëí=Ñ~áäëK==

854. q~åÖÉåí=mä~åÉ=qÜÉ=Éèì~íáçå=çÑ=íÜÉ=í~åÖÉåí=éä~åÉ=íç=íÜÉ=ëìêÑ~ÅÉ= ( )óIñÑò = =~í= ( )MMM òIóIñ =áë==

( )( ) ( )( )MMMóMMMñM óóóIñÑññóIñÑòò −+−=− K=

=

Page 235: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

225

855. kçêã~ä=íç=pìêÑ~ÅÉ=qÜÉ=Éèì~íáçå=çÑ=íÜÉ=åçêã~ä=íç=íÜÉ=ëìêÑ~ÅÉ= ( )óIñÑò = =~í=( )MMM òIóIñ =áë==

( ) ( ) N

òò

óIñÑ

óó

óIñÑ

ññ M

MMó

M

MMñ

M

−−

=−

=−

K=

===

8.9 Differential Operators =

råáí=îÉÅíçêë=~äçåÖ=íÜÉ=ÅççêÇáå~íÉ=~ñÉëW= ár

I= àr

I= âr

=pÅ~ä~ê=ÑìåÅíáçåë=EëÅ~ä~ê=ÑáÉäÇëFW= ( )òIóIñÑ I= ( )åON ñIIñIñì K =dê~ÇáÉåí=çÑ=~=ëÅ~ä~ê=ÑáÉäÇW= ìÖê~Ç I= ì∇ =

aáêÉÅíáçå~ä=ÇÉêáî~íáîÉW=ä

Ñ

∂∂

=

sÉÅíçê=ÑìåÅíáçå=EîÉÅíçê=ÑáÉäÇFW= ( )oInImcr

=

aáîÉêÖÉåÅÉ=çÑ=~=îÉÅíçê=ÑáÉäÇW= cÇáîr

I= cr⋅∇ =

`ìêä=çÑ=~=îÉÅíçê=ÑáÉäÇW= cÅìêär

I= cr

×∇ =

i~éä~Åá~å=çéÉê~íçêW= O∇ ===

856. dê~ÇáÉåí=çÑ=~=pÅ~ä~ê=cìåÅíáçå=

∂∂

∂∂

∂∂

=∇=ò

ÑI

ó

ÑI

ñ

ÑÑÑÖê~Ç I==

∂∂

∂∂

∂∂

=∇=åON ñ

ìII

ñ

ìI

ñ

ìììÖê~Ç K K=

=857. aáêÉÅíáçå~ä=aÉêáî~íáîÉ=

γ∂∂

+β∂∂

+α∂∂

=∂∂

Åçëò

ÑÅçë

ó

ÑÅçë

ñ

Ñ

ä

ÑI==

Page 236: 1300 Math Formulas

CHAPTER 8. DIFFERENTIAL CALCULUS

226

ïÜÉêÉ=íÜÉ=ÇáêÉÅíáçå=áë=ÇÉÑáåÉÇ=Äó=íÜÉ=îÉÅíçê=

( )γβα ÅçëIÅçëIÅçëär

I= NÅçëÅçëÅçë OOO =γ+β+α K===

858. aáîÉêÖÉåÅÉ=çÑ=~=sÉÅíçê=cáÉäÇ=

ò

o

ó

n

ñ

mccÇáî

∂∂

+∂∂

+∂∂

=⋅∇=rr

=

=859. `ìêä=çÑ=~=sÉÅíçê=cáÉäÇ=

onmñññ

âàá

ccÅìêä∂∂

∂∂

∂∂

=×∇=

rrr

rr=

âó

m

ñ

ñ

o

ò

ò

n

ó

o rrr

∂∂

−∂∂

+

∂∂

−∂∂

+

∂∂

−∂∂

= =

=860. i~éä~Åá~å=léÉê~íçê=

O

O

O

O

O

OO

ò

Ñ

ó

Ñ

ñ

ÑÑ

∂∂

+∂∂

+∂∂

=∇ =

=

861. ( ) ( ) MccÅìêäÇáî ≡×∇⋅∇=rr

==

862. ( ) ( ) MÑÑÖê~ÇÅìêä ≡∇×∇= ==

863. ( ) ( ) ÑÑÑÖê~ÇÇáî O∇=∇⋅∇= ==

864. ( ) ( ) ( ) ccccÇáîÖê~ÇcÅìêäÅìêä OOrrrrr

∇−⋅∇∇=∇−= ===

Page 237: 1300 Math Formulas

227

Chapter 9

Integral Calculus ====cìåÅíáçåëW=ÑI=ÖI=ìI=î=fåÇÉéÉåÇÉåí=î~êá~ÄäÉëW=ñI=íI=ξ =

fåÇÉÑáåáíÉ=áåíÉÖê~ä=çÑ=~=ÑìåÅíáçåW= ( )∫ ÇññÑ I= ( )∫ ÇññÖ I=£=

aÉêáî~íáîÉ=çÑ=~=ÑìåÅíáçåW= ( )ñó′ I= ( )ñÑ ′ I= ( )ñc′ I=£=oÉ~ä=Åçåëí~åíëW=`I=~I=ÄI=ÅI=ÇI=â=k~íìê~ä=åìãÄÉêëW=ãI=åI=áI=à===

9.1 Indefinite Integral =

865. ( ) ( ) `ñcÇññÑ +=∫ =áÑ= ( ) ( )ñÑñc =′ K=

=

866. ( )( ) ( )ñÑÇññÑ =′∫ =

=

867. ( ) ( )∫∫ = ÇññÑâÇññâÑ =

=

868. ( ) ( )[ ] ( ) ( )∫∫∫ +=+ ÇññÖÇññÑÇññÖñÑ =

=

869. ( ) ( )[ ] ( ) ( )∫∫∫ −=− ÇññÖÇññÑÇññÖñÑ =

=

870. ( ) ( ) `~ñc~

NÇñ~ñÑ +=∫ =

Page 238: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

228

871. ( ) ( ) `Ä~ñc~

NÇñÄ~ñÑ ++=+∫ =

=

872. ( ) ( ) ( ) `ñÑO

NÇññÑñÑ O +=′∫ =

=

873. ( )( ) ( ) `ñÑäåÇññÑ

ñÑ+=

′∫ =

=874. jÉíÜçÇ=çÑ=pìÄëíáíìíáçå=

( ) ( )( ) ( )∫∫ ′= ÇííìíìÑÇññÑ =áÑ= ( )íìñ = K=

=875. fåíÉÖê~íáçå=Äó=m~êíë=

∫∫ −= îÇììîìÇî I==

ïÜÉêÉ= ( )ñì I= ( )ñî =~êÉ=ÇáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåëK=====

9.2 Integrals of Rational Functions =

876. `~ñ~Çñ +=∫ =

=

877. `O

ññÇñ

O

+=∫ =

=

878. `P

ñÇññ

PO +=∫ =

=

879. `Né

ñÇññ

Néé +

+=

+

∫ I= Né −≠ K=

=

Page 239: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

229

880. ( ) ( )( ) `

Nå~

Ä~ñÇñÄ~ñ

Nåå +

++

=++

∫ I= Nå −≠ K=

=

881. `ñäåñ

Çñ+=∫ =

=

882. `Ä~ñäå~

N

Ä~ñ

Çñ++=

+∫ =

=

883. `ÇÅñäåÅ

~ÇÄÅñ

Å

~Çñ

ÇÅñ

Ä~ñO

++−

+=++

∫ =

=

884. ( )( ) `~ñ

Äñäå

Ä~

N

Äñ~ñ

Çñ+

++

−=

++∫ I= Ä~ ≠ K=

=

885. ( ) `Äñ~äå~Äñ~Ä

N

Äñ~

ñÇñO

++−+=+∫ =

=

886. ( ) ( ) `Äñ~äå~Äñ~~OÄñ~O

N

Ä

N

Äñ~

Çññ OO

P

O

+

+++−+=

+∫ =

=

887. ( ) `ñ

Äñ~äå

~

N

Äñ~ñ

Çñ+

+=

+∫ =

=

888. ( ) `ñ

Äñ~äå

~

Ä

N

Äñ~ñ

ÇñOO

++

+−=+∫ =

=

889. ( )

`Äñ~

~Äñ~äå

Ä

N

Äñ~

ñÇñOO +

+++=

+∫ =

=

Page 240: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

230

890. ( )

`Äñ~

~Äñ~äå~OÄñ~

Ä

N

Äñ~

Çññ O

PO

O

+

+

−+−+=+∫ =

=

891. ( ) ( ) `

ñ

Äñ~äå

~

N

Äñ~~

N

Äñ~ñ

ÇñOO

++

++

=+∫ =

=

892. `Nñ

Nñäå

O

N

ÇñO

++−

=−∫ =

=

893. `ñN

ñNäå

O

N

ñN

ÇñO

+−+

=−∫ =

=

894. `ñ~

ñ~äå

~O

N

ñ~

ÇñOO

+−+

=−∫ =

=

895. `~ñ

~ñäå

~O

N

ÇñOO

++−

=−∫ =

=

896. `ñí~åñN

Çñ N

O+=

+−∫ =

=

897. `~

ñí~å

~

N

ñ~

Çñ N

OO+=

+−∫ =

=

898. ( ) `~ñäåO

N

ñÇñ OO

OO++=

+∫ =

=

899. `~

Äñ~êÅí~å

N

Äñ~

ÇñO

+

=

+∫ I= M~Ä > K=

=

Page 241: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

231

900. `Ä

~ñäå

ÄO

N

Äñ~

ñÇñ O

O++=

+∫ =

=

901. ( ) `Äñ~

ñäå

~O

N

Äñ~ñ

ÇñO

O

O+

+=

+∫ =

=

902. `Äñ~

Äñ~äå

~ÄO

N

ñÄ~

ÇñOOO

+−+

=−∫ =

=

903. `~ÅQÄÄ~ñO

~ÅQÄÄ~ñOäå

~ÅQÄ

N

ÅÄñ~ñ

ÇñO

O

OO+

−++

−−+

−=

++∫ I=

M~ÅQÄO >− K==

904. `Ä~ÅQ

Ä~ñO~êÅí~å

Ä~ÅQ

O

ÅÄñ~ñ

ÇñOOO+

+

−=

++∫ I=

M~ÅQÄO <− K====

9.3 Integrals of Irrational Functions =

905. `Ä~ñ~

O

Ä~ñ

Çñ++=

+∫ =

=

906. ( ) `Ä~ñ~P

OÇñÄ~ñ O

P++=+∫ =

=

907. ( )`Ä~ñ

~P

ÄO~ñO

Ä~ñ

ñÇñO

++−

=+∫ =

=

Page 242: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

232

908. ( )( ) `Ä~ñ~NR

ÄO~ñPOÇñÄ~ññ O

P

O++

−=+∫ =

=

909. ( )

`~ÅÄÄ~ñ

~ÅÄÄ~ñäå

~ÅÄ

N

Ä~ñÅñ

Çñ+

−++−−+

−=

++∫ I==

M~ÅÄ >− K==

910. ( )

`Ä~Å

Ä~ñ~êÅí~å

Ä~Å

N

Ä~ñÅñ

Çñ+

−+

−=

++∫ I==

M~ÅÄ <− K==

911. ( )( ) −++=++

∫ ÇÅñÄ~ñÅ

NÇñ

ÇÅñ

Ä~ñ=

( ) ( ) `Ä~ñÅÇÅñ~äå~ÅÅ

ÄÅ~Ç++++

−− I= M~ > K=

=

912. ( )( ) −++=++

∫ ÇÅñÄ~ñÅ

NÇñ

ÇÅñ

Ä~ñ=

( )( ) `

Ä~ñÅ

ÇÅñ~~êÅí~å

~ÅÅ

ÄÅ~Ç+

++−

− I=E M~ < I= MÅ > FK==

=

913. ( ) ( ) `Äñ~ÄNMR

ñÄNR~ÄñNO~UOÇñÄñ~ñ P

P

OOOO ++

+−=+∫ =

=

914. ( )`Äñ~

ÄNR

ñÄP~ÄñQ~UO

Äñ~

ÇññP

OOOO

+++−

=+∫ =

=

915. `~Äñ~

~Äñ~äå

~

N

Äñ~ñ

Çñ+

++−+

=+∫ I= M~ > K=

=

Page 243: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

233

916. `~

Äñ~~êÅí~å

~

O

Äñ~ñ

Çñ+

−+

−=

+∫ I= M~ < K=

=

917. ( )( ) ( ) `Ä~

Äñ~êÅëáåÄ~ñÄñ~Çñ

ñÄ

ñ~+

++

+++−=+−

∫ =

=

918. ( )( ) ( ) `Ä~

ñÄ~êÅëáåÄ~ñÄñ~Çñ

ñÄ

ñ~+

+−

+−−+−=−+

∫ =

=

919. `ñ~êÅëáåñNÇññN

ñN O ++−−=−+

∫ =

=

920. ( )( )

`~Ä

~ñ~êÅëáåO

~Ä~ñ

Çñ+

−−

=−−∫ =

=

921. +−+−

=−+∫ OO ÅñÄñ~ÅQ

ÄÅñOÇñÅñÄñ~ =

`~ÅQÄ

ÄÅñO~êÅëáå

ÅU

~ÅQÄOP

O

++

−−+ =

=

922. ( ) `ÅÄñ~ñ~OÄ~ñOäå~

N

ÅÄñ~ñ

Çñ O

O+++++=

++∫ I==

M~ > K==

923. `~ÅQÄ~Q

Ä~ñO~êÅëáå

~

N

ÅÄñ~ñ

Çñ O

O+−

+−=

++∫ I= M~ < K=

=

924. `~ññäåO

~~ñ

O

ñÇñ~ñ OO

OOOOO +++++=+∫ =

=

Page 244: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

234

925. ( ) `~ñP

NÇñ~ññ O

POOOO ++=+∫ =

=

926. ( ) −++=+∫ OOOOOOO ~ñ~ñOU

ñÇñ~ññ =

`~ññäåU

~ OOQ

+++− =

=

927. `~ññäåñ

~ñÇñ

ñ

~ñ OOOO

O

OO

+++++

−=+

∫ =

=

928. `~ññäå~ñ

Çñ OO

OO+++=

+∫ =

=

929. `~ñ~

ñäå~~ñÇñ

ñ

~ñOO

OOOO

+++

++=+

∫ =

=

930. `~ñ~ñ

ñÇñ OO

OO++=

+∫ =

=

931. `~ññäåO

~~ñ

O

ñ

Çññ OOO

OO

OO

O

+++−+=+∫ =

=

932. `~ñ~

ñäå

~

N

~ññ

ÇñOOOO+

++=

+∫ =

=

933. `~ññäåO

~~ñ

O

ñÇñ~ñ OO

OOOOO +−+−−=−∫ =

=

934. ( ) `~ñP

NÇñ~ññ O

POOOO +−=−∫ =

Page 245: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

235

935. `ñ

~~êÅëáå~~ñÇñ

ñ

~ñ OOOO

++−=−

∫ =

=

936. `~ññäåñ

~ñÇñ

ñ

~ñ OOOO

O

OO

+−++−

−=−

∫ =

=

937. `~ññäå~ñ

Çñ OO

OO+−+=

−∫ =

=

938. `~ñ~ñ

ñÇñ OO

OO+−=

−∫ =

=

939. `~ññäåO

~~ñ

O

ñ

Çññ OOO

OO

OO

O

+−++−=−∫ =

=

940. `ñ

~~êÅëáå

~

N

~ññ

ÇñOO

+−=−∫ =

=

941. ( )

`~ñ

~

N

~ñ~ñ

ÇñOO

++−

=−+∫ =

=

942. ( )

`~ñ

~

N

~ñ~ñ

ÇñOO

+−+

−=−−∫ =

=

943. `ñ~

~ññ

ÇñO

OO

OOO+

−=

−∫ =

=

944. ( )

`~ñ~

ñ

ÇñOOOO

POO

+−

−=−

∫ =

=

Page 246: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

236

945. ( ) ( ) +−−−=−∫ OOOOOP

OO ~ñ~RñOU

ñÇñ~ñ =

`~ññäåU

~P OOQ

+−++ =

=

946. `~

ñ~êÅëáå

O

~ñ~

O

ñÇññ~

OOOOO ++−=−∫ =

=

947. ( ) `ñ~P

NÇññ~ñ O

POOOO +−−=−∫ =

=

948. ( ) `~

ñ~êÅëáå

U

~ñ~~ñO

U

ñÇññ~ñ

QOOOOOOO ++−−=−∫ =

=

949. `ñ~~

ñäå~ñ~Çñ

ñ

ñ~OO

OOOO

+−+

+−=−

∫ =

=

950. `~

ñ~êÅëáå

ñ

ñ~Çñ

ñ

ñ~ OO

O

OO

+−−

−=−

∫ =

=

951. `ñ~êÅëáåñN

ÇñO

+=−

∫ =

=

952. `~

ñëáå

ñ~

ÇñOO

+=−

∫ =

=

953. `ñ~ñ~

ñÇñ OO

OO+−−=

−∫ =

=

954. `~

ñ~êÅëáå

O

~ñ~

O

ñ

ñ~

Çññ OOO

OO

O

++−−=−∫ =

Page 247: 1300 Math Formulas

CHAPTER 9. INTEGRAL CALCULUS

237

955. ( )

`ñ~

ñ~

O

N

ñ~~ñ

ÇñOO

++−

−=−+∫ =

=

956. ( )

`ñ~

ñ~

O

N

ñ~~ñ

ÇñOO

+−+

−=−−∫ =

=

957. ( ) ( ) `

Äñ~

~Äñ~êÅëáå

N

ñ~Äñ

Çñ O

OOOO+

++

−=

−+∫ I= ~Ä > K=

=

958. ( )

I`Äñ~ñ~Ä~

Äñäå

Ä~

N

ñ~Äñ

ÇñOOOOOOOOO

+++−−

+

−=

−+∫ =

~Ä < K==

959. `ñ~

ñ~

ñ~ñ

ÇñO

OO

OOO+

−−=

−∫ =

=

960. ( ) ( ) `~

ñ~êÅëáå

U

~Pñ~ñO~R

U

ñÇññ~

QOOOOO

POO ++−−=−∫ =

=

961. ( )

`ñ~~

ñ

ñ~

ÇñOOOO

POO

+−

=−

∫ =

===

9.4 Integrals of Trigonometric Functions =

962. `ñÅçëñÇñëáå +−=∫ =

=

963. `ñëáåñÇñÅçë +=∫ =

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238

964. `ñOëáåQ

N

O

ñÇññëáåO +−=∫ =

=

965. `ñOëáåQ

N

O

ñÇññÅçëO ++=∫ =

=

966. `ñÅçëQ

PñPÅçë

NO

N`ñÅçëñÅçë

P

NÇññëáå PP +−=+−=∫ =

=

967. `ñëáåQ

PñPëáå

NO

N`ñëáå

P

NñëáåÇññÅçë PP ++=+−=∫ =

=

968. `O

ñí~åäåÇññÅëÅ

ñëáå

Çñ+== ∫∫ =

=

969. `QO

ñí~åäåÇññëÉÅ

ñÅçë

Çñ+

π

+== ∫∫ =

=

970. `ñÅçíÇññÅëÅñëáå

Çñ O

O+−== ∫∫ =

=

971. `ñí~åÇññëÉÅñÅçë

Çñ O

O+== ∫∫ =

=

972. `O

ñí~åäå

O

N

ñëáåO

ñÅçëÇññÅëÅ

ñëáå

ÇñO

P

P++−== ∫∫ =

=

973. `QO

ñí~åäå

O

N

ñÅçëO

ñëáåÇññëÉÅ

ñÅçë

ÇñO

P

P+

π

++== ∫∫ =

=

974. `ñOÅçëQ

NÇññÅçëñëáå +−=∫ =

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239

975. `ñëáåP

NÇññÅçëñëáå PO +=∫ =

=

976. `ñÅçëP

NÇññÅçëñëáå PO +−=∫ =

=

977. `ñQëáåPO

N

U

ñÇññÅçëñëáå OO +−=∫ =

=

978. `ñÅçëäåñÇñí~å +−=∫ =

=

979. `ñëÉÅ`ñÅçë

NÇñ

ñÅçë

ñëáåO

+=+=∫ =

=

980. `ñëáåQO

ñí~åäåÇñ

ñÅçë

ñëáåO

+−

π

+=∫ =

=

981. `ññí~åÇññí~åO +−=∫ =

=

982. `ñëáåäåñÇñÅçí +=∫ =

=

983. `ñÅëÅ`ñëáå

NÇñ

ñëáå

ñÅçëO

+−=+−=∫ =

=

984. `ñÅçëO

ñí~åäåÇñ

ñëáå

ñÅçëO

++=∫ =

=

985. `ññÅçíÇññÅçíO +−−=∫ =

=

986. `ñí~åäåñëáåñÅçë

Çñ+=∫ =

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240

987. `QO

ñí~åäå

ñëáå

N

ñÅçëñëáå

ÇñO

+

π

++−=∫ =

=

988. `O

ñí~åäå

ñÅçë

N

ñÅçëñëáå

ÇñO

++=∫ =

=

989. `ñÅçíñí~åñÅçëñëáå

ÇñOO

+−=∫ =

=

990. ( )( )

( )( ) `

åãO

ñåãëáå

åãO

ñåãëáåÇñåñëáåãñëáå +

−−

+++

−=∫ I=

OO åã ≠ K==

991. ( )( )

( )( ) `

åãO

ñåãÅçë

åãO

ñåãÅçëÇñåñÅçëãñëáå +

−−

−++

−=∫ I=

OO åã ≠ K==

992. ( )( )

( )( ) `

åãO

ñåãëáå

åãO

ñåãëáåÇñåñÅçëãñÅçë +

−−

+++

=∫ I=

OO åã ≠ K==

993. `ñëÉÅñÇñí~åñëÉÅ +=∫ =

=

994. `ñÅëÅñÇñÅçíñÅëÅ +−=∫ =

=

995. `Nå

ñÅçëÇññÅçëñëáå

Nåå +

+−=

+

∫ =

=

996. `Nå

ñëáåÇññÅçëñëáå

Nåå +

+=

+

∫ =

=

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241

997. `ñNñ~êÅëáåñÇññ~êÅëáå O +−+=∫ =

=

998. `ñNñ~êÅÅçëñÇññ~êÅÅçë O +−−=∫ =

=

999. ( ) `NñäåO

Nñ~êÅí~åñÇññ~êÅí~å O ++−=∫ =

=

1000. ( ) `NñäåO

NñÅçí~êÅñÇññÅçí~êÅ O +++=∫ =

===

9.5 Integrals of Hyperbolic Functions =

1001. `ñÅçëÜñÇñëáåÜ +=∫ =

=

1002. `ñëáåÜñÇñÅçëÜ +=∫ =

=

1003. `ñÅçëÜäåÇññí~åÜ +=∫ =

=

1004. `ñëáåÜäåÇññÅçíÜ +=∫ =

=

1005. `ñí~åÜñÇñëÉÅÜO +=∫ =

=

1006. `ñÅçíÜñÇñÅëÅÜO +−=∫ =

=

1007. `ñëÉÅÜñÇñí~åÜñëÉÅÜ +−=∫ =

=

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242

1008. `ñÅëÅÜñÇñÅçíÜñÅëÅÜ +−=∫ =

===

9.6 Integrals of Exponential and Logarithmic Functions

=

1009. `ÉÇñÉ ññ +=∫ =

=

1010. `~äå

~Çñ~

ññ +=∫ =

=

1011. `~

ÉÇñÉ

~ñ~ñ +=∫ =

=

1012. ( ) `N~ñ~

ÉÇññÉ

O

~ñ~ñ +−=∫ =

=

1013. `ññäåñÇññäå +−=∫ =

=

1014. `ñäåäåñäåñ

Çñ+=∫ =

=

1015. ( )

`Nå

N

ñäåñÇññäåñ O

Nåå +

+

−+

= +∫ =

=

1016. `ÉÄ~

ÄñÅçëÄÄñëáå~ÇñÄñëáåÉ ~ñ

OO

~ñ ++−

=∫ =

=

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243

1017. `ÉÄ~

ÄñëáåÄÄñÅçë~ÇñÄñÅçëÉ ~ñ

OO

~ñ +++

=∫ =

===

9.7 Reduction Formulas =

1018. ∫∫ −−= ÇñÉñã

åÉñ

ã

NÇñÉñ ãñNåãñåãñå =

=

1019. ( ) ∫∫ −− −

+−

−= Çññ

É

ã

ñNå

ÉÇñ

ñ

ÉNå

ãñ

ãñ

å

ãñ

I= Nå ≠ K=

=

1020. ∫∫ −− −−= ñÇñëáåÜ

å

NåñÅçëÜñëáåÜ

å

NñÇñëáåÜ OåNåå =

=

1021. ( ) ∫∫ −− −−

−−

−=ñëáåÜ

Çñ

ñëáåÜNå

ñÅçëÜ

ñëáåÜ

ÇñOåNåå

I= Nå ≠ K=

=

1022. ∫∫ −− −+= ñÇñÅçëÜ

å

NåñÅçëÜñÅçëÜñëáåÜ

å

NñÇñÅçëÜ OåNåå =

=

1023. ( ) ∫∫ −− −−

+−

−=ñÅçëÜ

Çñ

ñÅçëÜNå

ñëáåÜ

ñÅçëÜ

ÇñOåNåå

I= Nå ≠ K=

=

1024. ãå

ñÅçëÜñëáåÜñÇñÅçëÜñëáåÜ

NãNåãå

+=

−+

∫ =

∫ −

+−

+ ñÇñÅçëÜñëáåÜãå

Nã Oãå =

=

1025. ãå

ñÅçëÜñëáåÜñÇñÅçëÜñëáåÜ

NãNåãå

+=

+−

∫ =

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CHAPTER 9. INTEGRAL CALCULUS

244

∫ −

+−

− ñÇñÅçëÜñëáåÜãå

Nå ãOå =

=

1026. ∫∫ −− +−

−= ñÇñí~åÜñí~åÜNå

NñÇñí~åÜ OåNåå I= Nå ≠ K=

=

1027. ∫∫ −− +−

−= ñÇñÅçíÜñÅçíÜNå

NñÇñÅçíÜ OåNåå I= Nå ≠ K=

=

1028. ∫∫ −−

−−

+−

= ñÇñëÉÅÜNå

ñí~åÜñëÉÅÜñÇñëÉÅÜ Oå

Oåå I= Nå ≠ K=

=

1029. ∫∫ −− −+−= ñÇñëáå

å

NåñÅçëñëáå

å

NñÇñëáå OåNåå =

=

1030. ( ) ∫∫ −− −−

+−

−=ñëáå

Çñ

ñëáåNå

ñÅçë

ñëáå

ÇñOåNåå

I= Nå ≠ K=

=

1031. ∫∫ −− −+= ñÇñÅçë

å

NåñÅçëñëáå

å

NñÇñÅçë OåNåå =

=

1032. ( ) ∫∫ −− −−

+−

=ñÅçë

Çñ

ñÅçëNå

ñëáå

ñÅçë

ÇñOåNåå

I= Nå ≠ K=

=

1033. ãå

ñÅçëñëáåñÇñÅçëñëáå

NãNåãå

+=

−+

∫ =

∫ −

+−

+ ñÇñÅçëñëáåãå

Nã Oãå =

=

1034. ãå

ñÅçëñëáåñÇñÅçëñëáå

NãNåãå

+−=

+−

∫ =

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CHAPTER 9. INTEGRAL CALCULUS

245

∫ −

+−

+ ñÇñÅçëñëáåãå

Nå ãOå =

=

1035. ∫∫ −− −−

= ñÇñí~åñí~åNå

NñÇñí~å OåNåå I= Nå ≠ K=

=

1036. ∫∫ −− −−

−= ñÇñÅçíñÅçíNå

NñÇñÅçí OåNåå I= Nå ≠ K=

=

1037. ∫∫ −−

−−

+−

= ñÇñëÉÅNå

ñí~åñëÉÅñÇñëÉÅ Oå

Oåå I= Nå ≠ K=

=

1038. ∫∫ −−

−−

+−

−= ñÇñÅëÅNå

ñÅçíñÅëÅñÇñÅëÅ Oå

Oåå I= Nå ≠ K=

=

1039. ∫∫ −+

+−

+= ñÇñäåñ

ã

ñäåññÇñäåñ Nãå

ãNåãå =

=

1040. ( ) ∫∫

− −+

−−= Çñ

ñ

ñäå

ã

ñNå

ñäåÇñ

ñ

ñäåå

ã

å

ã

I= Nå ≠ K=

=

1041. ∫∫ −−= ñÇñäååñäåññÇñäå Nååå =

=

1042. ∫∫ −−= ñÇñÅçëÜñåñÅçëÜññÇñëáåÜñ Nååå =

=

1043. ∫∫ −−= ñÇñëáåÜñåñëáåÜññÇñÅçëÜñ Nååå =

=

1044. ∫∫ −+−= ñÇñÅçëñåñÅçëññÇñëáåñ Nååå =

=

1045. ∫∫ −−= ñÇñëáåñåñëáåññÇñÅçëñ Nååå =

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246

1046. ∫∫ −+−

+=

+−

+− Çñ

ñN

ñ

Nñëáå

ññÇñëáåñ

O

NåN

NåNå =

=

1047. ∫∫ −++

+=

+−

+− Çñ

ñN

ñ

NñÅçë

ññÇñÅçëñ

O

NåN

NåNå =

=

1048. ∫∫ ++−

+=

+−

+− Çñ

ñN

ñ

Nñí~å

ññÇñí~åñ

O

NåN

NåNå =

=

1049. ∫∫ +−=

+ Ä~ñ

Çñ

~

Ä

~

ñ

Ä~ñ

Çññåå

å

=

=

1050. ( ) ( )( )( ) NåOOåO ÅÄñ~ñ~ÅQÄNå

Ä~ñO

ÅÄñ~ñ

Çñ−

++−−

−−=

++∫ =

( )( )( ) ( )∫ −

++−−−

−NåOO

ÅÄñ~ñ

Çñ

~ÅQÄNå

~PåOOI= Nå ≠ K=

=

1051. ( ) ( ) ( ) ( ) ( ) I~ñ

Çñ

~NåO

PåO

~ñ~NåO

ñ

ÇñNåOOONåOOOåOO ∫∫ −−

+−−

++−

=+

Nå ≠ K==

1052. ( ) ( ) ( ) NåOOOåOO ~ñ~NåO

ñ

Çñ−

−−−=

−∫ =

( ) ( )∫ −−−

−−

NåOOO~ñ

Çñ

~NåO

PåOI= Nå ≠ K=

=====

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247

9.8 Definite Integral =

aÉÑáåáíÉ=áåíÉÖê~ä=çÑ=~=ÑìåÅíáçåW= ( )∫Ä

~

ÇññÑ I= ( )∫Ä

~

ÇññÖ I=£=

oáÉã~åå=ëìãW= ( )∑=

∆ξå

Nááá ñÑ ==

pã~ää=ÅÜ~åÖÉëW= áñ∆ ==^åíáÇÉêáî~íáîÉëW= ( )ñc I= ( )ñd ==iáãáíë=çÑ=áåíÉÖê~íáçåëW=~I=ÄI=ÅI=Ç==

=

1053. ( ) ( )∑∫=→∆

∞→∆ξ=

å

Nááá

Mñã~ñå

Ä

~

ñÑäáãÇññÑá

I==

ïÜÉêÉ== Nááá ñññ −−=∆ I== ááNá ññ ≤ξ≤− K===

==

Figure 179. =

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CHAPTER 9. INTEGRAL CALCULUS

248

1054. ~ÄÇñNÄ

~

−=∫ =

=

1055. ( ) ( )∫∫ =Ä

~

Ä

~

ÇññÑâÇññâÑ =

=

1056. ( ) ( )[ ] ( ) ( )∫∫∫ +=+Ä

~

Ä

~

Ä

~

ÇññÖÇññÑÇññÖñÑ =

=

1057. ( ) ( )[ ] ( ) ( )∫∫∫ −=−Ä

~

Ä

~

Ä

~

ÇññÖÇññÑÇññÖñÑ =

=

1058. ( ) MÇññÑ~

~

=∫ =

=

1059. ( ) ( )∫∫ −=~

Ä

Ä

~

ÇññÑÇññÑ =

=

1060. ( ) ( ) ( )∫∫∫ +=Ä

Å

Å

~

Ä

~

ÇññÑÇññÑÇññÑ =Ñçê= ÄÅ~ << K=

=

1061. ( ) MÇññÑÄ

~

≥∫ =áÑ= ( ) MñÑ ≥ =çå= [ ]ÄI~ K=

=

1062. ( ) MÇññÑÄ

~

≤∫ =áÑ= ( ) MñÑ ≤ =çå= [ ]ÄI~ K=

====

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249

1063. cìåÇ~ãÉåí~ä=qÜÉçêÉã=çÑ=`~äÅìäìë=

( ) ( ) ( ) ( )~cÄcñcÇññÑÄ

~

Ä

~

−==∫ =áÑ= ( ) ( )ñÑñc =′ K=

=1064. jÉíÜçÇ=çÑ=pìÄëíáíìíáçå==

fÑ= ( )íÖñ = I=íÜÉå==

( ) ( )( ) ( )∫∫ ′=Ç

Å

Ä

~

ÇííÖíÖÑÇññÑ I==

ïÜÉêÉ=( )~ÖÅ N−= I= ( )ÄÖÇ N−= K=

=1065. fåíÉÖê~íáçå=Äó=m~êíë=

( ) ∫∫ −=Ä

~

Ä

~

Ä

~

îÇììîìÇî =

=1066. qê~éÉòçáÇ~ä=oìäÉ=

( ) ( ) ( ) ( )

++

−= ∑∫

=

NááåM

Ä

~

ñÑOñÑñÑåO

~ÄÇññÑ =

=

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250

==

Figure 180. =

1067. páãéëçå∞ë=oìäÉ==

( ) ( ) ( ) ( ) ( )[ ++++−

=∫ PONM

Ä

~

ñÑQñÑOñÑQñÑåP

~ÄÇññÑ =

( ) ( ) ( )]åNåQ ñÑñÑQñÑO ++++ −K I==ïÜÉêÉ==

áå

~Ä~ñ á

−+= I= åIIOINIMá K= K==

=

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251

==

Figure 181. =

1068. ̂ êÉ~=råÇÉê=~=`ìêîÉ=

( ) ( ) ( )~cÄcÇññÑpÄ

~

−== ∫ I==

ïÜÉêÉ= ( ) ( )ñÑñc =′ K==

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252

==

Figure 182. =

1069. ̂ êÉ~=_ÉíïÉÉå=qïç=`ìêîÉë=

( ) ( )[ ] ( ) ( ) ( ) ( )~d~cÄdÄcÇññÖñÑpÄ

~

+−−=−= ∫ I==

ïÜÉêÉ= ( ) ( )ñÑñc =′ I= ( ) ( )ñÖñd =′ K==

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253

==

Figure 183. ===

9.9 Improper Integral =

1070. qÜÉ=ÇÉÑáåáíÉ=áåíÉÖê~ä== ( )∫Ä

~

ÇññÑ =áë=Å~ääÉÇ=~å=áãéêçéÉê=áåíÉÖê~ä=

áÑ==• ~=çê=Ä=áë=áåÑáåáíÉI=• ( )ñÑ ==Ü~ë==çåÉ==çê==ãçêÉ=éçáåíë=çÑ==ÇáëÅçåíáåìáíó======áå=íÜÉ=áåíÉêî~ä= [ ]ÄI~ K==

1071. fÑ= ( )ñÑ =áë=~=Åçåíáåìçìë=ÑìåÅíáçå=çå= [ )∞I~ I=íÜÉå==

( ) ( )∫∫ ∞→

~

ÇññÑäáãÇññÑ K=

=

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CHAPTER 9. INTEGRAL CALCULUS

254

==

Figure 184. =

1072. fÑ= ( )ñÑ =áë=~=Åçåíáåìçìë=ÑìåÅíáçå=çå= ( ]ÄI∞− I=íÜÉå==

( ) ( )∫∫ ∞−→∞−

åå

Ä

ÇññÑäáãÇññÑ K=

=

==

Figure 185.

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255

kçíÉ=W=qÜÉ=áãéêçéÉê=áåíÉÖê~äë=áå=NMTNI=NMTO=~êÉ=ÅçåîÉêÖÉåí=áÑ= íÜÉ= äáãáíë=Éñáëí=~åÇ=~êÉ=ÑáåáíÉX=çíÜÉêïáëÉ=íÜÉ= áåíÉÖê~äë=~êÉ=ÇáîÉêÖÉåíK==

1073. ( ) ( ) ( )∫∫∫∞

∞−

∞−

+=Å

Å

ÇññÑÇññÑÇññÑ =

=

==

Figure 186. =fÑ=Ñçê=ëçãÉ=êÉ~ä=åìãÄÉê=ÅI=ÄçíÜ=çÑ=íÜÉ=áåíÉÖê~äë=áå=íÜÉ=êáÖÜí=

ëáÇÉ= ~êÉ= ÅçåîÉêÖÉåíI= íÜÉå= íÜÉ= áåíÉÖê~ä= ( )∫∞

∞−

ÇññÑ = áë= ~äëç=====

ÅçåîÉêÖÉåíX=çíÜÉêïáëÉ=áí=áë=ÇáîÉêÖÉåíK==

1074. ̀ çãé~êáëçå=qÜÉçêÉãë=iÉí== ( )ñÑ =~åÇ== ( )ñÖ ==ÄÉ==Åçåíáåìçìë==ÑìåÅíáçåë==çå=íÜÉ=ÅäçëÉÇ=áåíÉêî~ä= [ )∞I~ K= pìééçëÉ= íÜ~í= ( ) ( )ñÑñÖM ≤≤ = Ñçê= ~ää= ñ= áå=[ )∞I~ K=

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256

• fÑ= ( )∫∞

~

ÇññÑ =áë=ÅçåîÉêÖÉåíI=íÜÉå= ( )∫∞

~

ÇññÖ =áë=~äëç=

=====ÅçåîÉêÖÉåíI=

• fÑ= ( )∫∞

~

ÇññÖ =áë=ÇáîÉêÖÉåíI=íÜÉå= ( )∫∞

~

ÇññÑ =áë=~äëç=ÇáîÉêÖÉåíK=

=1075. ̂ ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=

=

fÑ= ( )∫∞

~

ÇññÑ =áë=ÅçåîÉêÖÉåíI=íÜÉå=íÜÉ=áåíÉÖê~ä ( )∫∞

~

ÇññÑ =áë=~Äëç-

äìíÉäó=ÅçåîÉêÖÉåíK====

1076. aáëÅçåíáåìçìë=fåíÉÖê~åÇ=iÉí= ( )ñÑ =ÄÉ=~=ÑìåÅíáçå=ïÜáÅÜ=áë=Åçåíáåìçìë=çå=íÜÉ=áåíÉêî~ä==[ )ÄI~ =Äìí=áë=ÇáëÅçåíáåìçìë=~í= Äñ = K=qÜÉå==

( ) ( )∫∫ε−

+→ε=

Ä

~M

Ä

~

ÇññÑäáãÇññÑ =

=

==

Figure 187.

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CHAPTER 9. INTEGRAL CALCULUS

257

1077. iÉí= ( )ñÑ =ÄÉ=~=Åçåíáåìçìë=ÑìåÅíáçå=Ñçê=~ää=êÉ~ä=åìãÄÉêë==ñ==áå=íÜÉ=áåíÉêî~ä== [ ]ÄI~ ==ÉñÅÉéí==Ñçê==ëçãÉ=éçáåí==Å==áå= ( )ÄI~ K=qÜÉå=

( ) ( ) ( )∫∫∫δ+

+→δ

ε−

+→ε+=

Ä

ÅM

Å

~M

Ä

~

ÇññÑäáãÇññÑäáãÇññÑ K=

=

==

Figure 188. ===

9.10 Double Integral =cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= ( )óIñÑ I= ( )îIìÑ I=£=

açìÄäÉ=áåíÉÖê~äëW= ( )∫∫o

ÇñÇóóIñÑ I= ( )∫∫o

ÇñÇóóIñÖ I=£=

oáÉã~åå=ëìãW= ( )∑∑= =

∆∆ã

å

Nààáàá óñîIìÑ =

pã~ää=ÅÜ~åÖÉëW= áñ∆ I= àó∆ =

oÉÖáçåë=çÑ=áåíÉÖê~íáçåW=oI=p==mçä~ê=ÅççêÇáå~íÉëW= ê I=θ =

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258

^êÉ~W=^=pìêÑ~ÅÉ=~êÉ~W=p=sçäìãÉ=çÑ=~=ëçäáÇW=s=j~ëë=çÑ=~=ä~ãáå~W=ã=aÉåëáíóW= ( )óIñρ =cáêëí=ãçãÉåíëW= ñj I= ój =

jçãÉåíë=çÑ=áåÉêíá~W= ñf I= óf I= Mf =

`Ü~êÖÉ=çÑ=~=éä~íÉW=n=`Ü~êÖÉ=ÇÉåëáíóW= ( )óIñσ =`ççêÇáå~íÉë=çÑ=ÅÉåíÉê=çÑ=ã~ëëW= ñ I= ó =^îÉê~ÖÉ=çÑ=~=ÑìåÅíáçåW=µ ==

1078. aÉÑáåáíáçå=çÑ=açìÄäÉ=fåíÉÖê~ä=qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=êÉÅí~åÖäÉ= [ ] [ ]ÇIÅÄI~ × =áë=ÇÉÑáåÉÇ=íç=ÄÉ==

( )[ ] [ ]

( )∑∑∫∫= =→∆

→∆×

∆∆=ã

å

Nààáàá

Móã~ñMñã~ñ

ÇIÅÄI~

óñîIìÑäáãÇ^óIñÑà

á

I==

ïÜÉêÉ= ( )àá îIì =áë=ëçãÉ=éçáåí=áå=íÜÉ=êÉÅí~åÖäÉ=

( ) ( )àNàáNá óIóñIñ −− × I=~åÇ= Nááá ñññ −−=∆ I= Nààà óóó −−=∆ K=

=

=== =Figure 189.

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259

qÜÉ=ÇçìÄäÉ=áåíÉÖê~ä=çîÉê=~=ÖÉåÉê~ä=êÉÖáçå=o=áë==

( ) ( )[ ] [ ]∫∫∫∫×

=ÇIÅÄI~o

Ç^óIñÖÇ^óIñÑ I==

ïÜÉêÉ=êÉÅí~åÖäÉ= [ ] [ ]ÇIÅÄI~ × =Åçåí~áåë=oI==( ) ( )óIñÑóIñÖ = =áÑ= ( )óIñÑ =áë=áå=o=~åÇ= ( ) MóIñÖ = =çíÜÉêïáëÉK=

=

==

Figure 190. =

1079. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫ +=+ooo

Ç^óIñÖÇ^óIñÑÇ^óIñÖóIñÑ =

=

1080. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫ −=−ooo

Ç^óIñÖÇ^óIñÑÇ^óIñÖóIñÑ =

=

1081. ( ) ( )∫∫∫∫ =oo

Ç^óIñÑâÇ^óIñâÑ I==

ïÜÉêÉ=â=áë=~=Åçåëí~åíK==

1082. fÑ= ( ) ( )óIñÖóIñÑ ≤ =çå=oI=íÜÉå= ( ) ( )∫∫∫∫ ≤oo

Ç^óIñÖÇ^óIñÑ K=

=1083. fÑ= ( ) MóIñÑ ≥ =çå=o=~åÇ= op⊂ I=íÜÉå=

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260

= ( ) ( )∫∫∫∫ ≤op

Ç^óIñÑÇ^óIñÑ K=

=

==

Figure 191. =

1084. fÑ= ( ) MóIñÑ ≥ =çå=o=~åÇ=o=~åÇ=p=~êÉ=åçå-çîÉêä~ééáåÖ=

êÉÖáçåëI=íÜÉå= ( ) ( ) ( )∫∫∫∫∫∫ +=∪ popo

Ç^óIñÑÇ^óIñÑÇ^óIñÑ K==

eÉêÉ= po∪ =áë=íÜÉ=ìåáçå=çÑ=íÜÉ=êÉÖáçåë=o=~åÇ=pK==

==

Figure 192. ==

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261

1085. fíÉê~íÉÇ=fåíÉÖê~äë=~åÇ=cìÄáåá∞ë=qÜÉçêÉã=

( ) ( )( )

( )

∫ ∫∫∫ =Ä

~

ñè

ñéo

ÇóÇñóIñÑÇ^óIñÑ ==

Ñçê=~=êÉÖáçå=çÑ=íóéÉ=fI==( ) ( ) ( ){ }ñèóñéIÄñ~öóIño ≤≤≤≤= K=

=

==

Figure 193. =

( ) ( )( )

( )

∫ ∫∫∫ =Ç

Å

óî

óìo

ÇñÇóóIñÑÇ^óIñÑ ==

Ñçê=~=êÉÖáçå=çÑ=íóéÉ=ffI=( ) ( ) ( ){ }ÇóÅIóîñóìöóIño ≤≤≤≤= K=

=

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262

==

Figure 194. =

1086. açìÄäÉ=fåíÉÖê~äë=çîÉê=oÉÅí~åÖìä~ê=oÉÖáçåë==fÑ=o=áë=íÜÉ=êÉÅí~åÖìä~ê=êÉÖáçå= [ ] [ ]ÇIÅÄI~ × I=íÜÉå==

( ) ( ) ( )∫ ∫∫ ∫∫∫

=

=

Ç

Å

Ä

~

Ä

~

Ç

Åo

ÇóÇñóIñÑÇñÇóóIñÑÇñÇóóIñÑ K==

=få=íÜÉ=ëéÉÅá~ä=Å~ëÉ=ïÜÉêÉ=íÜÉ=áåíÉÖê~åÇ= ( )óIñÑ =Å~å=ÄÉ=ïêáí-íÉå=~ë= ( ) ( )óÜñÖ =ïÉ=Ü~îÉ==

( ) ( ) ( ) ( ) ( )

== ∫∫∫∫∫∫

Ç

Å

Ä

~oo

ÇóóÜÇññÖÇñÇóóÜñÖÇñÇóóIñÑ K==

=1087. ̀ Ü~åÖÉ=çÑ=s~êá~ÄäÉë=

( ) ( ) ( )[ ] ( )( )ÇìÇî

îIì

óIñîIìóIîIìñÑÇñÇóóIñÑ

po ∂∂

= ∫∫∫∫ I==

ïÜÉêÉ=( )( ) M

î

ó

ì

óî

ñ

ì

ñ

îIì

óIñ≠

∂∂

∂∂

∂∂

∂∂

=∂∂

áë= íÜÉ= à~ÅçÄá~å= çÑ= íÜÉ= íê~åë-

Ñçêã~íáçåë= ( ) ( )îIìóIñ → I=~åÇ=p=áë=íÜÉ=éìääÄ~Åâ=çÑ=o=ïÜáÅÜ=

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263

Å~å=ÄÉ=ÅçãéìíÉÇ=Äó= ( )îIìññ = I= ( )îIìóó = =áåíç=íÜÉ=ÇÉÑáåá-íáçå=çÑ=oK===

1088. mçä~ê=`ççêÇáå~íÉë=θ= Åçëêñ I= θ= ëáåêó K==

=

==

Figure 195. =

1089. açìÄäÉ=fåíÉÖê~äë=áå=mçä~ê=`ççêÇáå~íÉë==qÜÉ=aáÑÑÉêÉåíá~ä=ÇñÇó=Ñçê=mçä~ê=`ççêÇáå~íÉë=áë==

( )( ) θ=θθ∂

∂= êÇêÇÇêÇ

óIñÇñÇó K==

=iÉí=íÜÉ=êÉÖáçå=o=áë=ÇÉíÉêãáåÉÇ=~ë=ÑçääçïëW=

( ) ( )θ≤≤θ≤ ÜêÖM I= β≤θ≤α I=ïÜÉêÉ= π≤α−β O K==qÜÉå==

( ) ( )( )

( )

∫ ∫∫∫β

α

θ

θ

θθθ=Ü

Öo

êÇêÇëáåêIÅçëêÑÇñÇóóIñÑ K=

=

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264

==

Figure 196. =fÑ=íÜÉ=êÉÖáçå=o=áë=íÜÉ=éçä~ê=êÉÅí~åÖäÉ=ÖáîÉå=Äó==

Äê~M ≤≤≤ I= β≤θ≤α I=ïÜÉêÉ= π≤α−β O I===íÜÉå==

( ) ( )∫ ∫∫∫β

α

θθθ=Ä

~o

êÇêÇëáåêIÅçëêÑÇñÇóóIñÑ K==

=

==

Figure 197. ==

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265

1090. ̂ êÉ~=çÑ=~=oÉÖáçå=

( )

( )

∫ ∫=Ä

~

ñÑ

ñÖ

ÇóÇñ^ =EÑçê=~=íóéÉ=f=êÉÖáçåFK=

=

==

Figure 198. =

( )

( )

∫ ∫=Ç

Å

óè

óé

ÇñÇó^ =EÑçê=~=íóéÉ=ff=êÉÖáçåFK=

=

==

Figure 199. ==

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266

1091. sçäìãÉ=çÑ=~=pçäáÇ=

( )∫∫=o

Ç^óIñÑs K==

=

==

Figure 200. =fÑ=o=áë=~=íóéÉ=f=êÉÖáçå=ÄçìåÇÉÇ=Äó= ~ñ = I= Äñ = I= ( )ñÜó = I=

( )ñÖó = I=íÜÉå==

( ) ( )( )

( )

∫ ∫∫∫ ==Ä

~

ñÖ

ñÜo

ÇóÇñóIñÑÇ^óIñÑs K==

=fÑ=o=áë=~=íóéÉ=ff=êÉÖáçå=ÄçìåÇÉÇ=Äó= Åó = I= Çó = I= ( )óèñ = I=

( )óéñ = I=íÜÉå=

( ) ( )( )

( )

∫ ∫∫∫ ==Ç

Å

óè

óéo

ÇñÇóóIñÑÇ^óIñÑs K==

=

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267

fÑ== ( ) ( )óIñÖóIñÑ ≥ ==çîÉê==~==êÉÖáçå==oI==íÜÉå==íÜÉ==îçäìãÉ==çÑ=íÜÉ= ëçäáÇ= ÄÉíïÉÉå= ( )óIñÑòN = = ~åÇ= ( )óIñÖòO = = çîÉê= o= áë=ÖáîÉå=Äó=

( ) ( )[ ]∫∫ −=o

Ç^óIñÖóIñÑs K==

=1092. ̂ êÉ~=~åÇ=sçäìãÉ=áå=mçä~ê=`ççêÇáå~íÉë=

fÑ=p=áë=~=êÉÖáçå=áå=íÜÉ=ñó-éä~åÉ=ÄçìåÇÉÇ=Äó= α=θ I= β=θ I=( )θ= Üê I= ( )θ= Öê I==

íÜÉå==

( )

( )

∫ ∫∫∫β

α

θ

θ

θ==Ö

Üp

êÇêÇÇ^^ I==

( )∫∫ θθ=p

êÇêÇIêÑs K==

=

==

Figure 201. =

1093. pìêÑ~ÅÉ=^êÉ~=

ÇñÇóó

ò

ñ

òNp

o

OO

∫∫

∂∂

+

∂∂

+= =

=

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268

1094. j~ëë=çÑ=~=i~ãáå~=

( )∫∫ρ=o

Ç^óIñã I==

ïÜÉêÉ==íÜÉ==ä~ãáå~==çÅÅìéáÉë==~==êÉÖáçå==o=~åÇ==áíë==ÇÉåëáíó==~í=~=éçáåí=EñIóF=áë= ( )óIñρ K====

1095. jçãÉåíë=qÜÉ=ãçãÉåí=çÑ=íÜÉ=ä~ãáå~=~Äçìí=íÜÉ==ñ-~ñáë==áë=ÖáîÉå=Äó=Ñçê-ãìä~=

( )∫∫ ρ=o

ñ Ç^óIñój K==

=qÜÉ=ãçãÉåí=çÑ=íÜÉ=ä~ãáå~=~Äçìí=íÜÉ=ó-~ñáë=áë=

( )∫∫ ρ=o

ó Ç^óIññj K=

=qÜÉ=ãçãÉåí=çÑ=áåÉêíá~=~Äçìí=íÜÉ=ñ-~ñáë=áë=

( )∫∫ ρ=o

Oñ Ç^óIñóf K==

=qÜÉ=ãçãÉåí=çÑ=áåÉêíá~=~Äçìí=íÜÉ=ó-~ñáë=áë=

( )∫∫ ρ=o

Oó Ç^óIññf K==

=qÜÉ=éçä~ê=ãçãÉåí=çÑ=áåÉêíá~=áë=

( ) ( )∫∫ ρ+=o

OOM Ç^óIñóñf K==

=1096. ̀ ÉåíÉê=çÑ=j~ëë=

( )( )

( )∫∫

∫∫∫∫ ρ

ρ=ρ==

o

o

o

ó

Ç^óIñ

Ç^óIññ

Ç^óIññã

N

ã

jñ I==

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269

( )( )

( )∫∫

∫∫∫∫ ρ

ρ=ρ==

o

o

o

ñ

Ç^óIñ

Ç^óIñó

Ç^óIñóã

N

ã

jó K==

=1097. ̀ Ü~êÖÉ=çÑ=~=mä~íÉ=

( )∫∫σ=o

Ç^óIñn I==

ïÜÉêÉ=ÉäÉÅíêáÅ~ä=ÅÜ~êÖÉ=áë=ÇáëíêáÄìíÉÇ=çîÉê=~=êÉÖáçå=o=~åÇ=áíë=ÅÜ~êÖÉ=ÇÉåëáíó=~í=~=éçáåí=EñIóF=áë= ( )óIñσ K====

1098. ̂ îÉê~ÖÉ=çÑ=~=cìåÅíáçå=

( )∫∫=µo

Ç^óIñÑp

NI==

ïÜÉêÉ= ∫∫=o

Ç^p K==

===

9.11 Triple Integral =cìåÅíáçåë=çÑ=íÜêÉÉ=î~êá~ÄäÉëW= ( )òIóIñÑ I= ( )òIóIñÖ I=£=

qêáéäÉ=áåíÉÖê~äëW= ( )∫∫∫d

ÇsòIóIñÑ I= ( )∫∫∫d

ÇsòIóIñÖ I=£=

oáÉã~åå=ëìãW= ( )∑∑∑= = =

∆∆∆ã

å

é

Nââàáâàá òóñïIîIìÑ =

pã~ää=ÅÜ~åÖÉëW= áñ∆ I= àó∆ I= âò∆ =

iáãáíë=çÑ=áåíÉÖê~íáçåW=~I=ÄI=ÅI=ÇI=êI=ë=oÉÖáçåë=çÑ=áåíÉÖê~íáçåW=dI=qI=p==`óäáåÇêáÅ~ä=ÅççêÇáå~íÉëW= ê I=θ I=ò=péÜÉêáÅ~ä=ÅççêÇáå~íÉëW= ê I=θ I=ϕ =sçäìãÉ=çÑ=~=ëçäáÇW=s=

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270

j~ëë=çÑ=~=ëçäáÇW=ã==aÉåëáíóW= ( )òIóIñµ =`ççêÇáå~íÉë=çÑ=ÅÉåíÉê=çÑ=ã~ëëW= ñ I= ó I= ò =cáêëí=ãçãÉåíëW= ñój I= óòj I= ñòj =

jçãÉåíë=çÑ=áåÉêíá~W= ñóf I= óòf I= ñòf I= ñf I= óf I= òf I= Mf =

==

1099. aÉÑáåáíáçå=çÑ=qêáéäÉ=fåíÉÖê~ä=qÜÉ=íêáéäÉ=áåíÉÖê~ä=çîÉê=~=é~ê~ääÉäÉéáéÉÇ= [ ] [ ] [ ]ëIêÇIÅÄI~ ×× =áë=ÇÉÑáåÉÇ=íç=ÄÉ==

( )[ ] [ ] [ ]

( ) IòóñïIîIìÑäáãÇsòIóIñÑã

å

é

Nââàáâàá

Mòã~ñMóã~ñMñã~ñ

ëIêÇIÅÄI~

â

à

á∑∑∑∫∫∫= = =

→∆→∆→∆

××

∆∆∆=

ïÜÉêÉ= ( )âàá ïIîIì =áë=ëçãÉ=éçáåí=áå=íÜÉ=é~ê~ääÉäÉéáéÉÇ=

( ) ( ) ( )âNâàNàáNá òIòóIóñIñ −−− ×× I=~åÇ= Nááá ñññ −−=∆ I=

Nààà óóó −−=∆ I= Nâââ òòò −−=∆ K=

=

1100. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫∫∫∫ +=+ddd

ÇsòIóIñÖÇsòIóIñÑÇsòIóIñÖòIóIñÑ

=

1101. ( ) ( )[ ] ( ) ( )∫∫∫∫∫∫∫∫∫ −=−ddd

ÇsòIóIñÖÇsòIóIñÑÇsòIóIñÖòIóIñÑ

=

1102. ( ) ( )∫∫∫∫∫∫ =dd

ÇsòIóIñÑâÇsòIóIñâÑ I==

ïÜÉêÉ=â=áë=~=Åçåëí~åíK==

1103. fÑ== ( ) MòIóIñÑ ≥ =~åÇ=d=~åÇ=q=~êÉ=åçåçîÉêä~ééáåÖ=Ä~ëáÅ=êÉÖáçåëI=íÜÉå==

( ) ( ) ( )∫∫∫∫∫∫∫∫∫ +=∪ qdqd

ÇsòIóIñÑÇsòIóIñÑÇsòIóIñÑ K==

eÉêÉ= qd∪ =áë=íÜÉ=ìåáçå=çÑ=íÜÉ=êÉÖáçåë=d=~åÇ=qK==

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271

1104. bî~äì~íáçå=çÑ=qêáéäÉ=fåíÉÖê~äë=Äó=oÉéÉ~íÉÇ=fåíÉÖê~äë=fÑ=íÜÉ=ëçäáÇ=d=áë=íÜÉ=ëÉí=çÑ=éçáåíë= ( )òIóIñ =ëìÅÜ=íÜ~í== ( ) ( ) ( )óIñòóIñIoóIñ ON χ≤≤χ∈ I=íÜÉå==

= ( ) ( )( )

( )

ÇñÇóÇòòIóIñÑÇñÇóÇòòIóIñÑo

óIñ

óIñd

O

N

∫∫ ∫∫∫∫

=

χ

χ

I==

ïÜÉêÉ=o=áë=éêçàÉÅíáçå=çÑ=d=çåíç=íÜÉ=ñó-éä~åÉK==fÑ=íÜÉ=ëçäáÇ=d=áë=íÜÉ=ëÉí=çÑ=éçáåíë= ( )òIóIñ =ëìÅÜ=íÜ~í=

( ) ( ) ( ) ( )óIñòóIñIñóñIÄñ~ ONON χ≤≤χϕ≤≤ϕ≤≤ I=íÜÉå==

( ) ( )( )

( )

( )

( )

ÇñÇóÇòòIóIñÑÇñÇóÇòòIóIñÑÄ

~

ñ

ñ

óIñ

óIñd

O

N

O

N

∫ ∫ ∫∫∫∫

=

ϕ

ϕ

χ

χ

==

=1105. qêáéäÉ=fåíÉÖê~äë=çîÉê=m~ê~ääÉäÉéáéÉÇ=

fÑ=d=áë=~=é~ê~ääÉäÉéáéÉÇ= [ ] [ ] [ ]ëIêÇIÅÄI~ ×× I=íÜÉå=

( ) ( ) ÇñÇóÇòòIóIñÑÇñÇóÇòòIóIñÑÄ

~

Ç

Å

ë

êd∫ ∫ ∫∫∫∫

= K==

=få= = íÜÉ=ëéÉÅá~ä=Å~ëÉ= =ïÜÉêÉ= íÜÉ= áåíÉÖê~åÇ= = ( )òIóIñÑ = =Å~å=ÄÉ=ïêáííÉå=~ë= ( ) ( ) ( )òâóÜñÖ =ïÉ=Ü~îÉ==

( ) ( ) ( ) ( )

= ∫∫∫∫∫∫

ë

ê

Ç

Å

Ä

~d

ÇòòâÇóóÜÇññÖÇñÇóÇòòIóIñÑ K==

=1106. ̀ Ü~åÖÉ=çÑ=s~êá~ÄäÉë=

( ) =∫∫∫d

ÇñÇóÇòòIóIñÑ =

( ) ( ) ( )[ ] ( )( ) IÇñÇóÇò

ïIîIì

òIóIñïIîIìòIïIîIìóIïIîIìñÑ

p∫∫∫ ∂

∂= =

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272

ïÜÉêÉ= =( )( ) M

ï

ò

î

ò

ì

òï

ó

î

ó

ì

óï

ñ

î

ñ

ì

ñ

ïIîIì

òIóIñ≠

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

=∂∂

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M−×=ε =áë=éÉêãáííáîáíó=çÑ=ÑêÉÉ=ëé~ÅÉK==

==

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294

Chapter 10

Differential Equations ====cìåÅíáçåë=çÑ=çåÉ=î~êá~ÄäÉW=óI=éI=èI=ìI=ÖI=ÜI=dI=eI=êI=ò==^êÖìãÉåíë=EáåÇÉéÉåÇÉåí=î~êá~ÄäÉëFW=ñI=ó=cìåÅíáçåë=çÑ=íïç=î~êá~ÄäÉëW= ( )óIñÑ I= ( )óIñj I= ( )óIñk =

cáêëí=çêÇÉê=ÇÉêáî~íáîÉW= ó′ I= ì′ I= ó& I=Çí

ÇóI=£=

pÉÅçåÇ=çêÇÉê=ÇÉêáî~íáîÉëW= ó ′′ I= ó&& I=O

O

Çí

fÇI=£=

m~êíá~ä=ÇÉêáî~íáîÉëW=í

ì

∂∂

I=O

O

ñ

ì

∂∂

I=£=

k~íìê~ä=åìãÄÉêW=å=m~êíáÅìä~ê=ëçäìíáçåëW= Nó I= éó =

oÉ~ä=åìãÄÉêëW=âI=íI=`I= N` I= O` I=éI=èI=α I=β =oççíë=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåëW= Nλ I= Oλ =qáãÉW=í=qÉãéÉê~íìêÉW=qI=p=mçéìä~íáçå=ÑìåÅíáçåW= ( )ím ==j~ëë=çÑ=~å=çÄàÉÅíW=ã=píáÑÑåÉëë=çÑ=~=ëéêáåÖW=â=aáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãW=ó=^ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW=^=cêÉèìÉåÅóW=ω =a~ãéáåÖ=ÅçÉÑÑáÅáÉåíW= γ =mÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíW=δ =^åÖìä~ê=Çáëéä~ÅÉãÉåíW=θ =mÉåÇìäìã=äÉåÖíÜW=i=

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^ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóW=Ö=`ìêêÉåíW=f=oÉëáëí~åÅÉW=o=fåÇìÅí~åÅÉW=i=`~é~Åáí~åÅÉW=`===

10.1 First Order Ordinary Differential Equations

=1164. iáåÉ~ê=bèì~íáçåë=

( ) ( )ñèóñéÇñ

Çó=+ K==

=qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

( ) ( )( )ñì

`Çññèñìó

+= ∫ I==

ïÜÉêÉ==

( ) ( )( )∫= ÇññéÉñéñì K=

=1165. pÉé~ê~ÄäÉ=bèì~íáçåë=

( ) ( ) ( )óÜñÖóIñÑÇñ

Çó== =

=qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó=

( ) ( ) `ÇññÖóÜ

Çó+= ∫∫ I==

çê=( ) ( ) `ñdóe += K=

===

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296

1166. eçãçÖÉåÉçìë=bèì~íáçåë=

qÜÉ= ÇáÑÑÉêÉåíá~ä= Éèì~íáçå= ( )óIñÑÇñ

Çó= = áë= ÜçãçÖÉåÉçìëI= áÑ=

íÜÉ=ÑìåÅíáçå= ( )óIñÑ =áë=ÜçãçÖÉåÉçìëI=íÜ~í=áë==( ) ( )óIñÑíóIíñÑ = K==

=

qÜÉ=ëìÄëíáíìíáçå=ñ

óò = =EíÜÉå= òñó = F=äÉ~Çë=íç=íÜÉ=ëÉé~ê~ÄäÉ=

Éèì~íáçå=

( )òINÑòÇñ

Çòñ =+ K==

=1167. _Éêåçìääá=bèì~íáçå=

( ) ( ) åóñèóñéÇñ

Çó=+ K==

=qÜÉ=ëìÄëíáíìíáçå= åNóò −= =äÉ~Çë=íç=íÜÉ=äáåÉ~ê=Éèì~íáçå==

( ) ( ) ( ) ( )ñèåNòñéåNÇñ

Çò−=−+ K==

=1168. oáÅÅ~íá=bèì~íáçå=

( ) ( ) ( ) OóñêóñèñéÇñ

Çó++= =

=fÑ=~=é~êíáÅìä~ê=ëçäìíáçå= Nó =áë=âåçïåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäì-íáçå=Å~å=ÄÉ=çÄí~áåÉÇ=ïáíÜ=íÜÉ=ÜÉäé=çÑ=ëìÄëíáíìíáçå=

Nóó

−= I=ïÜáÅÜ=äÉ~Çë=íç=íÜÉ=Ñáêëí=çêÇÉê=äáåÉ~ê=Éèì~íáçå==

( ) ( )[ ] ( )ñêòñêóOñèÇñ

ÇòN −+−= K==

===

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297

1169. bñ~Åí=~åÇ=kçåÉñ~Åí=bèì~íáçåë=qÜÉ=Éèì~íáçå==( ) ( ) MÇóóIñkÇñóIñj =+ ==

áë=Å~ääÉÇ=Éñ~Åí=áÑ==

ñ

k

ó

j

∂∂

=∂∂

I==

~åÇ=åçåÉñ~Åí=çíÜÉêïáëÉK==qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

( ) ( ) `ÇóóIñkÇñóIñj =+ ∫∫ K==

=1170. o~Çáç~ÅíáîÉ=aÉÅ~ó=

âóÇí

Çó−= I==

ïÜÉêÉ= ( )íó =áë=íÜÉ=~ãçìåí=çÑ=ê~Çáç~ÅíáîÉ=ÉäÉãÉåí=~í=íáãÉ=íI=â=áë=íÜÉ=ê~íÉ=çÑ=ÇÉÅ~óK===qÜÉ=ëçäìíáçå=áë==( ) âí

MÉóíó −= I=ïÜÉêÉ= ( )MóóM = =áë=íÜÉ=áåáíá~ä=~ãçìåíK=

=1171. kÉïíçå∞ë=i~ï=çÑ=`ççäáåÖ=

( )pqâÇí

Çq−−= I==

ïÜÉêÉ= ( )íq =áë=íÜÉ=íÉãéÉê~íìêÉ=çÑ=~å=çÄàÉÅí=~í=íáãÉ=íI=p=áë=íÜÉ=

íÉãéÉê~íìêÉ= çÑ= íÜÉ= ëìêêçìåÇáåÖ= ÉåîáêçåãÉåíI= â= áë= ~= éçëá-íáîÉ=Åçåëí~åíK===qÜÉ=ëçäìíáçå=áë=( ) ( ) âí

M Épqpíq −−+= I==

ïÜÉêÉ= ( )MqqM = = áë= íÜÉ= áåáíá~ä= íÉãéÉê~íìêÉ= çÑ= íÜÉ= çÄàÉÅí= ~í=

íáãÉ= Mí = K====

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298

1172. mçéìä~íáçå=aóå~ãáÅë=EiçÖáëíáÅ=jçÇÉäF=

−=

j

mNâm

Çí

ÇmI==

ïÜÉêÉ= ( )ím =áë=éçéìä~íáçå=~í=íáãÉ=íI=â=áë=~=éçëáíáîÉ=Åçåëí~åíI=j=áë=~=äáãáíáåÖ=ëáòÉ=Ñçê=íÜÉ=éçéìä~íáçåK===qÜÉ=ëçäìíáçå=çÑ=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áë==

( ) ( ) âíMM

M

Émjm

jmím −−+= I=ïÜÉêÉ= ( )MmmM = =áë=íÜÉ=áåáíá~ä=éçéì-

ä~íáçå=~í=íáãÉ= Mí = K======

10.2 Second Order Ordinary Differential Equations

=1173. eçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=`çÉÑÑáÅáÉåíë==

Mèóóéó =+′+′′ K==qÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçå=áë==

MèéO =+λ+λ K===fÑ= Nλ = ~åÇ= Oλ = ~êÉ= ÇáëíáåÅí= êÉ~ä= êççíë= çÑ= íÜÉ= ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåI=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

ñO

ñN

ON É`É`ó λλ += I=ïÜÉêÉ==

N` =~åÇ= O` =~êÉ=áåíÉÖê~íáçå=Åçåëí~åíëK====

fÑ=O

éON −=λ=λ I=íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

( ) ñO

é

ON Éñ``ó−

+= K===fÑ= Nλ =~åÇ= Oλ =~êÉ=ÅçãéäÉñ=åìãÄÉêëW=

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299

áN β+α=λ I= áO β−α=λ I=ïÜÉêÉ==

O

é−=α I=

O

éèQ O−=β I==

íÜÉå=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==( )ñëáå`ñÅçë`Éó ON

ñ β+β= α K===

1174. fåÜçãçÖÉåÉçìë=iáåÉ~ê=bèì~íáçåë=ïáíÜ=`çåëí~åí=======================`çÉÑÑáÅáÉåíë==

( )ñÑèóóéó =+′+′′ K===qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë=ÖáîÉå=Äó==

Üé óóó += I=ïÜÉêÉ==

éó =áë==~=é~êíáÅìä~ê==ëçäìíáçå=çÑ==íÜÉ=áåÜçãçÖÉåÉçìë=Éèì~íáçå=

~åÇ= Üó =áë=íÜÉ=ÖÉåÉê~ä=ëçäìíáçå=çÑ=íÜÉ=~ëëçÅá~íÉÇ=ÜçãçÖÉåÉ-çìë=Éèì~íáçå=EëÉÉ=íÜÉ=éêÉîáçìë=íçéáÅ=NNTPFK==fÑ=íÜÉ=êáÖÜí=ëáÇÉ=Ü~ë=íÜÉ=Ñçêã==( ) ( ) ( )( )ñëáåñmñÅçëñmÉñÑ NN

ñ β+β= α I==íÜÉå=íÜÉ==é~êíáÅìä~ê=ëçäìíáçå= éó =áë=ÖáîÉå=Äó==

( ) ( )( )ñëáåñoñÅçëñoÉñó ONñâ

é β+β= α I==

ïÜÉêÉ= íÜÉ=éçäóåçãá~äë= ( )ñoN =~åÇ= ( )ñoO =Ü~îÉ= íç=ÄÉ= ÑçìåÇ=Äó=ìëáåÖ=íÜÉ=ãÉíÜçÇ=çÑ=ìåÇÉíÉêãáåÉÇ=ÅçÉÑÑáÅáÉåíëK==• fÑ= áβ+α =áë=åçí=~=êççí=çÑ=íÜÉ=ÅÜ~ê~ÅíÉêáëíáÅ=Éèì~íáçåI=íÜÉå=

íÜÉ=éçïÉê= Mâ = I=• fÑ= áβ+α =áë=~=ëáãéäÉ=êççíI=íÜÉå= Nâ = I=• fÑ= áβ+α =áë=~=ÇçìÄäÉ=êççíI=íÜÉå= Oâ = K===

1175. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ó=jáëëáåÖ=( )óIñÑó ′=′′ K==

pÉí= óì ′= K=qÜÉå=íÜÉ=åÉï=Éèì~íáçå=ë~íáëÑáÉÇ=Äó=î=áë==( )ìIñÑì =′ I==

ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK=

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300

1176. aáÑÑÉêÉåíá~ä=bèì~íáçåë=ïáíÜ=ñ=jáëëáåÖ=( )óIóÑó ′=′′ K==

pÉí= óì ′= K=páåÅÉ==

Çó

Çìì

Çñ

Çó

Çó

Çì

Çñ

Çìó ===′′ I==

ïÉ=Ü~îÉ=

( )ìIóÑÇó

Çìì = I==

ïÜáÅÜ=áë=~=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK===

1177. cêÉÉ=råÇ~ãéÉÇ=sáÄê~íáçåë=qÜÉ=ãçíáçå=çÑ=~=j~ëë=çå=~=péêáåÖ=áë=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=Éèì~-íáçå==

Mâóóã =+&& I==ïÜÉêÉ==ã=áë=íÜÉ=ã~ëë=çÑ=íÜÉ=çÄàÉÅíI=â=áë=íÜÉ=ëíáÑÑåÉëë=çÑ=íÜÉ=ëéêáåÖI=ó=áë=Çáëéä~ÅÉãÉåí=çÑ=íÜÉ=ã~ëë=Ñêçã=ÉèìáäáÄêáìãK==qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==

( )δ−ω= íÅçë^ó M I==

ïÜÉêÉ==^=áë=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíI=

Mω =áë=íÜÉ=ÑìåÇ~ãÉåí~ä=ÑêÉèìÉåÅóI=íÜÉ=éÉêáçÇ=áë=M

Oq

ωπ

= I=

δ =áë=éÜ~ëÉ=~åÖäÉ=çÑ=íÜÉ=Çáëéä~ÅÉãÉåíK=qÜáë=áë=~å=Éñ~ãéäÉ=çÑ=ëáãéäÉ=Ü~êãçåáÅ=ãçíáçåK===

1178. cêÉÉ=a~ãéÉÇ=sáÄê~íáçåë=Mâóóóã =+γ+ &&& I=ïÜÉêÉ==

γ =áë=íÜÉ=Ç~ãéáåÖ=ÅçÉÑÑáÅáÉåíK==qÜÉêÉ=~êÉ=P=Å~ëÉë=Ñçê=íÜÉ=ÖÉåÉê~ä=ëçäìíáçåW==

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301

`~ëÉ=NK âãQO >γ =EçîÉêÇ~ãéÉÇF=

( ) íí ON _É^Éíó λλ += I==ïÜÉêÉ==

ãO

âãQO

N

−γ−γ−=λ I=

ãO

âãQO

O

−γ+γ−=λ K==

=`~ëÉ=OK= âãQO =γ EÅêáíáÅ~ääó=Ç~ãéÉÇF=

( ) ( ) íÉ_í^íó λ+= I==ïÜÉêÉ==

ãO

γ−=λ K=

=`~ëÉ=PK= âãQO <γ =EìåÇÉêÇ~ãéÉÇF==

( ) ( )δ−ω=γ

−íÅçë^Éíó

íãO I=ïÜÉêÉ==

OâãQ γ−=ω K==

=1179. páãéäÉ=mÉåÇìäìã=

Mi

Ö

Çí

ÇO

O

=θ+θ

I=

ïÜÉêÉ= θ = áë= íÜÉ= ~åÖìä~ê= Çáëéä~ÅÉãÉåíI= i= áë= íÜÉ= éÉåÇìäìã=äÉåÖíÜI=Ö=áë=íÜÉ=~ÅÅÉäÉê~íáçå=çÑ=Öê~îáíóK==qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=Ñçê=ëã~ää=~åÖäÉë=θ =áë==

( ) íi

Öëáåí ã~ñθ=θ I=íÜÉ=éÉêáçÇ=áë=

Ö

iOq π= K==

=1180. oi`=`áêÅìáí=

( ) ( )íÅçëbísf`

N

Çí

Çfo

Çí

fÇi MO

O

ωω=′=++ I=

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302

ïÜÉêÉ=f= áë=íÜÉ=ÅìêêÉåí=áå=~å=oi`=ÅáêÅìáí=ïáíÜ=~å=~Å=îçäí~ÖÉ=ëçìêÅÉ= ( ) ( )íëáåbís M ω= K===

=qÜÉ=ÖÉåÉê~ä=ëçäìíáçå=áë==( ) ( )ϕ−ω++= íëáå^É`É`íf íê

Oíê

NON I=

ïÜÉêÉ==

iO`

iQoo

ê

O

OIN

−±−= I==

OOO

O

M

o`

Ni

b^

ω+

−ω

ω= I==

ω−

ω=ϕ

o`

N

o

i~êÅí~å I==

N` I= O` =~êÉ=Åçåëí~åíë=ÇÉéÉåÇáåÖ=çå=áåáíá~ä=ÅçåÇáíáçåëK====

10.3. Some Partial Differential Equations =

1181. qÜÉ=i~éä~ÅÉ=bèì~íáçå=

ì

ñ

ìO

O

O

O

=∂∂

+∂∂

=

~ééäáÉë=íç=éçíÉåíá~ä=ÉåÉêÖó=ÑìåÅíáçå= ( )óIñì ==Ñçê==~= =ÅçåëÉê-î~íáîÉ= ÑçêÅÉ= ÑáÉäÇ= áå= íÜÉ= ñó-éä~åÉK= m~êíá~ä= ÇáÑÑÉêÉåíá~ä= Éèì~-íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=ÉääáéíáÅK===

1182. qÜÉ=eÉ~í=bèì~íáçå=

í

ì

ó

ì

ñ

ìO

O

O

O

∂∂

=∂∂

+∂∂

=

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303

~ééäáÉë= íç= íÜÉ= íÉãéÉê~íìêÉ= ÇáëíêáÄìíáçå= ( )óIñì = áå= íÜÉ= ñó-éä~åÉ=ïÜÉå=ÜÉ~í=áë=~ääçïÉÇ=íç=Ñäçï=Ñêçã=ï~êã=~êÉ~ë=íç=Åççä=çåÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜáë=íóéÉ=~êÉ=Å~ääÉÇ=é~ê~ÄçäáÅK===

1183. qÜÉ=t~îÉ=bèì~íáçå=

O

O

O

O

O

O

í

ì

ó

ì

ñ

ì

∂∂

=∂∂

+∂∂

=

~ééäáÉë=íç=íÜÉ=Çáëéä~ÅÉãÉåí= ( )óIñì =çÑ=îáÄê~íáåÖ=ãÉãÄê~åÉë=~åÇ= çíÜÉê= ï~îÉ= ÑìåÅíáçåëK= qÜÉ= Éèì~íáçåë= çÑ= íÜáë= íóéÉ= ~êÉ=Å~ääÉÇ=ÜóéÉêÄçäáÅK====

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304

Chapter 11

Series ====

11.1 Arithmetic Series =fåáíá~ä=íÉêãW= N~ =kíÜ=íÉêãW= å~ =aáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=ëìÅÅÉëëáîÉ=íÉêãëW=Ç=kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= åp ===

1184. ( )ÇNå~ÇO~Ç~~ NOåNåå −+==+=+= −− K ==

1185. áNåáNåOåN ~~~~~~ −+− +==+=+ K ==

1186. O

~~~ NáNá

á+− +

= =

=

1187. ( )å

O

ÇNå~Oå

O

~~p NåN

å ⋅−+

=⋅+

= =

=====

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CHAPTER 11. SERIES

305

11.2 Geometric Series =fåáíá~ä=íÉêãW= N~ =kíÜ=íÉêãW= å~ =`çããçå=ê~íáçW=è=kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å=pìã=çÑ=íÜÉ=Ñáêëí=å=íÉêãëW= åp =pìã=íç=áåÑáåáíóW=p===

1188. NåNNåå è~è~~ −

− == ==

1189. áNåáNåOåN ~~~~~~ −+− ⋅==⋅=⋅ K ==

1190. NáNáá ~~~ +− ⋅= =

=

1191. ( )Nè

Nè~

~è~p

åNNå

å −−

=−−

= =

=

1192. èN

~päáãp N

åå −

==∞→

=

cçê= Nè < I=íÜÉ=ëìã=p=ÅçåîÉêÖÉë=~ë= ∞→å K=

===

11.3 Some Finite Series =kìãÄÉê=çÑ=íÉêãë=áå=íÜÉ=ëÉêáÉëW=å===

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CHAPTER 11. SERIES

306

1193. ( )O

NåååPON

+=++++ K =

=1194. ( )NåååOSQO +=++++ K =

=1195. ( ) OåNåORPN =−++++ K =

=

1196. ( ) ( ) ( ) ( )O

NåâOåNåâOâNââ

−+=−+++++++ K =

=

1197. ( )( )S

NåONåååPON OOOO ++

=++++ K =

=

1198. ( ) OPPPP

O

NåååPON

+

=++++ K =

=

1199. ( ) ( )P

NåQåNåORPN

OOOOO −=−++++ K =

=

1200. ( ) ( )NåOåNåORPN OOPPPP −=−++++ K ==

1201. OO

N

U

N

Q

N

O

NN

å=++++++ KK =

=

1202. ( ) NNåå

N

QP

N

PO

N

ON

N=+

+++

⋅+

⋅+

⋅KK =

=

1203. ( ) É>Nå

N

>P

N

>O

N

>N

NN =+

−+++++ KK =

===

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307

11.4 Infinite Series =pÉèìÉåÅÉW={ }å~ =cáêëí=íÉêãW= N~ =kíÜ=íÉêãW= å~ ===

1204. fåÑáåáíÉ=pÉêáÉë=

KK ++++=∑∞

=åON

Nåå ~~~~ =

=1205. kíÜ=m~êíá~ä=pìã=

åON

å

Nååå ~~~~p +++==∑

=

K =

=1206. ̀ çåîÉêÖÉåÅÉ=çÑ=fåÑáåáíÉ=pÉêáÉë=

i~Nå

å =∑∞

=

I=áÑ= ipäáã åå

=∞→

=

=1207. kíÜ=qÉêã=qÉëí=

• fÑ=íÜÉ=ëÉêáÉë=∑∞

=Nåå~ áë=ÅçåîÉêÖÉåíI=íÜÉå= M~äáã å

å=

∞→K==

• fÑ= M~äáã åå

≠∞→

I=íÜÉå=íÜÉ=ëÉêáÉë=áë=ÇáîÉêÖÉåíK=

===

11.5 Properties of Convergent Series =

`çåîÉêÖÉåí=pÉêáÉëW= ^~Nå

å =∑∞

=

I= _ÄNå

å =∑∞

=

=

oÉ~ä=åìãÄÉêW=Å=

Page 318: 1300 Math Formulas

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308

1208. ( ) _^Ä~Ä~Nå

åNå

åNå

åå +=+=+ ∑∑∑∞

=

=

=

=

=

1209. Å^~ÅÅ~Nå

åNå

å == ∑∑∞

=

=

K==

===

11.6 Convergence Tests =

1210. qÜÉ=`çãé~êáëçå=qÉëí=

iÉí=∑∞

=Nåå~ =~åÇ=∑

=NååÄ =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= åå Ä~M ≤< =Ñçê=~ää=åK==

• fÑ=∑∞

=NååÄ áë=ÅçåîÉêÖÉåí=íÜÉå=∑

=Nåå~ áë=~äëç=ÅçåîÉêÖÉåíK==

• fÑ=∑∞

=Nåå~ áë=ÇáîÉêÖÉåí=íÜÉå=∑

=NååÄ áë=~äëç=ÇáîÉêÖÉåíK=

=1211. qÜÉ=iáãáí=`çãé~êáëçå=qÉëí=

iÉí=∑∞

=Nåå~ =~åÇ=∑

=NååÄ =ÄÉ=ëÉêáÉë=ëìÅÜ=íÜ~í= å~ =~åÇ= åÄ =~êÉ=éçëá-

íáîÉ=Ñçê=~ää=åK==

• fÑ= ∞<<∞→

å

å

å Ä

~äáãM = íÜÉå= ∑

=Nåå~ = ~åÇ= ∑

=NååÄ ~êÉ= ÉáíÜÉê= ÄçíÜ=

ÅçåîÉêÖÉåí=çê=ÄçíÜ=ÇáîÉêÖÉåíK=

• fÑ= MÄ

~äáã

å

å

å=

∞→=íÜÉå=∑

=NååÄ ÅçåîÉêÖÉåí=áãéäáÉë=íÜ~í=∑

=Nåå~ =áë=

~äëç=ÅçåîÉêÖÉåíK=

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309

• fÑ= ∞=∞→

å

å

å Ä

~äáã =íÜÉå=∑

=NååÄ ÇáîÉêÖÉåí=áãéäáÉë=íÜ~í=∑

=Nåå~ =áë=

~äëç=ÇáîÉêÖÉåíK==

1212. é-ëÉêáÉë=

é-ëÉêáÉë=∑∞

=Nåéå

N=ÅçåîÉêÖÉë=Ñçê= Né > =~åÇ=ÇáîÉêÖÉë=Ñçê=

NéM ≤< K==

1213. qÜÉ=fåíÉÖê~ä=qÉëí=iÉí= ( )ñÑ = ÄÉ= ~= ÑìåÅíáçå= ïÜáÅÜ= áë= ÅçåíáåìçìëI= éçëáíáîÉI= ~åÇ=ÇÉÅêÉ~ëáåÖ=Ñçê=~ää= Nñ ≥ K=qÜÉ=ëÉêáÉë==

( ) ( ) ( ) ( ) ( ) KK +++++=∑∞

=

åÑPÑOÑNÑåÑNå

=

ÅçåîÉêÖÉë=áÑ= ( )∫∞

N

ÇññÑ ÅçåîÉêÖÉëI=~åÇ=ÇáîÉêÖÉë=áÑ=

( ) ∞→∫å

N

ÇññÑ =~ë= ∞→å K=

=1214. qÜÉ=o~íáç=qÉëí=

iÉí=∑∞

=Nåå~ =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=

• fÑ= N~

~äáã

å

å<+

∞→=íÜÉå=∑

=Nåå~ áë=ÅçåîÉêÖÉåíK=

• fÑ= N~

~äáã

å

å>+

∞→=íÜÉå=∑

=Nåå~ =áë=ÇáîÉêÖÉåíK=

• fÑ= N~

~äáã

å

å=+

∞→= íÜÉå=∑

=Nåå~ =ã~ó=ÅçåîÉêÖÉ=çê=ÇáîÉêÖÉ=~åÇ=

íÜÉ= ê~íáç= íÉëí= áë= áåÅçåÅäìëáîÉX= ëçãÉ= çíÜÉê= íÉëíë= ãìëí= ÄÉ=ìëÉÇK==

=

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310

1215. qÜÉ=oççí=qÉëí=

iÉí=∑∞

=Nåå~ =ÄÉ=~=ëÉêáÉë=ïáíÜ=éçëáíáîÉ=íÉêãëK=

• fÑ= N~äáã åå

å<

∞→=íÜÉå=∑

=Nåå~ =áë=ÅçåîÉêÖÉåíK=

• fÑ= N~äáã åå

å>

∞→=íÜÉå=∑

=Nåå~ =áë=ÇáîÉêÖÉåíK=

• fÑ= N~äáã åå

å=

∞→= íÜÉå=∑

=Nåå~ =ã~ó=ÅçåîÉêÖÉ=çê=ÇáîÉêÖÉI=Äìí=

åç=ÅçåÅäìëáçå=Å~å=ÄÉ=Çê~ïå=Ñêçã=íÜáë=íÉëíK====

11.7 Alternating Series =

1216. qÜÉ=^äíÉêå~íáåÖ=pÉêáÉë=qÉëí=EiÉáÄåáò∞ë=qÜÉçêÉãF==iÉí={ }å~ =ÄÉ=~=ëÉèìÉåÅÉ=çÑ=éçëáíáîÉ=åìãÄÉêë=ëìÅÜ=íÜ~í=

åNå ~~ <+ =Ñçê=~ää=åK=M~äáã å

å=

∞→K==

qÜÉå= íÜÉ= ~äíÉêå~íáåÖ= ëÉêáÉë= ( )∑∞

=

−Nå

åå ~N = ~åÇ= ( )∑

=

−−Nå

åNå ~N =

ÄçíÜ=ÅçåîÉêÖÉK=====

1217. ̂ ÄëçäìíÉ=`çåîÉêÖÉåÅÉ=

• ^= ëÉêáÉë= ∑∞

=Nåå~ = áë= ~ÄëçäìíÉäó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë=

∑∞

=Nåå~ =áë=ÅçåîÉêÖÉåíK==

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311

• fÑ=íÜÉ=ëÉêáÉë=∑∞

=Nåå~ áë=~ÄëçäìíÉäó=ÅçåîÉêÖÉåí=íÜÉå=áí=áë=Åçå-

îÉêÖÉåíK==

1218. ̀ çåÇáíáçå~ä=`çåîÉêÖÉåÅÉ=

^= ëÉêáÉë= ∑∞

=Nåå~ áë= ÅçåÇáíáçå~ääó= ÅçåîÉêÖÉåí= áÑ= íÜÉ= ëÉêáÉë= áë=

ÅçåîÉêÖÉåí=Äìí=áë=åçí=~ÄëçäìíÉäó=ÅçåîÉêÖÉåíK====

11.8 Power Series =oÉ~ä=åìãÄÉêëW=ñI= Mñ =

mçïÉê=ëÉêáÉëW=∑∞

=Må

ååñ~ I= ( )∑

=

−Må

åMå ññ~ =

tÜçäÉ=åìãÄÉêW=å=o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o==

1219. mçïÉê=pÉêáÉë=áå=ñ=

KK +++++=∑∞

=

åå

OONM

åå ñ~ñ~ñ~~ñ~ =

=1220. mçïÉê=pÉêáÉë=áå= ( )Mññ − =

( ) ( ) ( ) ( ) KK +−++−+−+=−∑∞

=

åMå

OMOMNM

åMå ññ~ññ~ññ~~ññ~

=1221. fåíÉêî~ä=çÑ=`çåîÉêÖÉåÅÉ===

qÜÉ=ëÉí=çÑ=íÜçëÉ=î~äìÉë=çÑ=ñ=Ñçê=ïÜáÅÜ=íÜÉ=ÑìåÅíáçå=

( ) ( )∑∞

=

−=Må

åMå ññ~ñÑ =áë=ÅçåîÉêÖÉåí=áë=Å~ääÉÇ==íÜÉ==áåíÉêî~ä=çÑ=

ÅçåîÉêÖÉåÅÉK=

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312

1222. o~Çáìë=çÑ=`çåîÉêÖÉåÅÉ=fÑ=íÜÉ=áåíÉêî~ä=çÑ=ÅçåîÉêÖÉåÅÉ=áë== ( )oñIoñ MM +− ==Ñçê==ëçãÉ=

Mo ≥ I=íÜÉ=o=áë=Å~ääÉÇ==íÜÉ=ê~Çáìë=çÑ=ÅçåîÉêÖÉåÅÉK==fí=áë=ÖáîÉå=~ë=

åå

å ~

Näáão

∞→= =çê=

å

å ~

~äáão

+∞→

= K==

===

11.9 Differentiation and Integration of Power Series

=`çåíáåìçìë=ÑìåÅíáçåW= ( )ñÑ =

mçïÉê=ëÉêáÉëW=∑∞

=Må

ååñ~ =

tÜçäÉ=åìãÄÉêW=å=o~Çáìë=çÑ=`çåîÉêÖÉåÅÉW=o===

1223. aáÑÑÉêÉåíá~íáçå=çÑ=mçïÉê=pÉêáÉë=

iÉí= ( ) K+++==∑∞

=

OONM

åå ñ~ñ~~ñ~ñÑ =Ñçê= oñ < K==

qÜÉåI= = Ñçê= oñ < I= ( )ñÑ = áë= ÅçåíáåìçìëI= íÜÉ= ÇÉêáî~íáîÉ= ( )ñÑ ′ =

Éñáëíë=~åÇ=

( ) K+++=′ OONM ñ~

Çñ

Çñ~

Çñ

Ç~

Çñ

ÇñÑ =

∑∞

=

−=+++=Nå

Nåå

OPON ñå~ñ~Pñ~O~ K K=

===

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CHAPTER 11. SERIES

313

1224. fåíÉÖê~íáçå=çÑ=mçïÉê=pÉêáÉë=

iÉí= ( ) K+++==∑∞

=

OONM

åå ñ~ñ~~ñ~ñÑ =Ñçê= oñ < K==

qÜÉåI==Ñçê= oñ < I=íÜÉ=áåÇÉÑáåáíÉ=áåíÉÖê~ä= ( )∫ ÇññÑ Éñáëíë=~åÇ==

( ) K+++= ∫∫∫∫ Çññ~ñÇñ~Çñ~ÇññÑ OONM =

`Nå

ñ~

P

ñ~

O

ñ~ñ~

å

P

O

O

NM ++

=+++= ∑∞

=

+

K K=

===

11.10 Taylor and Maclaurin Series =tÜçäÉ=åìãÄÉêW=å=aáÑÑÉêÉåíá~ÄäÉ=ÑìåÅíáçåW= ( )ñÑ =oÉã~áåÇÉê=íÉêãW= åo ===

1225. q~óäçê=pÉêáÉë=

( ) ( )( )( ) ( ) ( )( ) ( )( )K+

−′′+−′+=

−=∑

= >O

~ñ~Ñ~ñ~Ñ~Ñ

~ñ~ÑñÑ

O

åå

= =( ) ( )( )

å

åå

o>å

~ñ~Ñ+

−+ K==

=1226. qÜÉ=oÉã~áåÇÉê=^ÑíÉê=åHN=qÉêãë=áë=ÖáîÉå=Äó==

( )( )( )( )>Nå

~ñÑo

NåNå

å +−ξ

=++

I=== ñ~ <ξ< K=

=1227. j~Åä~ìêáå=pÉêáÉë=

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314

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )å

ååO

åå o

ñMÑ

>O

ñMÑñMÑMÑ

ñMÑñÑ +++

′′+′+==∑

=

K

====

11.11 Power Series Expansions for Some Functions

=tÜçäÉ=åìãÄÉêW=å=oÉ~ä=åìãÄÉêW=ñ===

1228. KK ++++++=>å

ñ

>P

ñ

>O

ññNÉ

åPOñ =

=

1229. ( ) ( ) ( )KK ++++++=

~äåñ

>P

~äåñ

>O

~äåñ

>N

~äåñN~

åPOñ =

=

1230. ( ) ( )KK ±

+−

++−+−=++

ñN

Q

ñ

P

ñ

O

ñññNäå

NååQPO

I= NñN ≤<− K=

=

1231.

++++=

−+

KT

ñ

R

ñ

P

ññO

ñN

ñNäå

TRP

I= Nñ < K=

=

1232.

+−

+

+−

++−

= KRP

R

N

P

N

NñOñäå I= Mñ > K=

=

1233. ( )( ) KK ±−

++−+−=>åO

ñN

>S

ñ

>Q

ñ

>O

ñNñÅçë

åOåSQO

=

=

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CHAPTER 11. SERIES

315

1234. ( )( ) KK ±

+−

++−+−=+

>NåO

ñN

>T

ñ

>R

ñ

>P

ñññëáå

NåOåTRP

=

=

1235. K+++++=OUPR

ñSO

PNR

ñNT

NR

ñO

P

ñññí~å

VTRP

I=O

ñπ

< K=

=

1236.

++++−= K

QTOR

ñO

VQR

ñO

QR

ñ

P

ñ

ñ

NñÅçí

TRP

I= π<ñ K=

=

1237. ( )( )( ) K

K

KK +

+⋅⋅−⋅⋅

++⋅⋅

⋅+

⋅+=

+

NåOåOSQO

ñNåORPN

RQO

ñPN

PO

ñññ~êÅëáå

NåORP

I=

Nñ < K=

=

1238. ( )( )( ) I

NåOåOSQO

ñNåORPN

RQO

ñPN

PO

ññ

Oñ~êÅÅçë

NåORP

+

+⋅⋅−⋅⋅

++⋅⋅

⋅+

⋅+−

π=

+

KK

KK

Nñ < K=

=

1239. ( )KK ±

+−

++−+−=+

NåO

ñN

T

ñ

R

ñ

P

ñññ~êÅí~å

NåOåTRP

I= Nñ ≤ K=

=

1240. ( ) KK ++++++=>åO

ñ

>S

ñ

>Q

ñ

>O

ñNñÅçëÜ

åOSQO

=

=

1241. ( ) KK ++

+++++=+

>NåO

ñ

>T

ñ

>R

ñ

>P

ñññëáåÜ

NåOTRP

=

=====

Page 326: 1300 Math Formulas

CHAPTER 11. SERIES

316

11.12 Binomial Series =tÜçäÉ=åìãÄÉêëW=åI=ã=oÉ~ä=åìãÄÉêW=ñ=`çãÄáå~íáçåëW= ã

å ` ===

1242. ( ) åãå

ãOO

åN

åå ññ`ñ`ñ`NñN ++++++=+ KK ==

1243. ( ) ( )[ ]>ã

NãåNåå`ã

å −−−=

KI= Nñ < K=

=

1244. K+−+−=+

PO ñññNñN

NI= Nñ < K=

=

1245. K++++=−

PO ñññNñN

NI= Nñ < K=

=

1246. K+⋅⋅⋅

⋅⋅−

⋅⋅⋅

+⋅

−+=+USQO

ñRPN

SQO

ñPN

QO

ñ

O

ñNñN

QPO

I= Nñ ≤ K=

=

1247. K+⋅⋅⋅⋅⋅⋅

−⋅⋅⋅⋅

+⋅⋅

−+=+NOVSP

ñURON

VSP

ñRON

SP

ñON

P

ñNñN

QPOP I= Nñ ≤ K=

===

11.13 Fourier Series =fåíÉÖê~ÄäÉ=ÑìåÅíáçåW= ( )ñÑ =cçìêáÉê=ÅçÉÑÑáÅáÉåíëW= M~ I= å~ I= åÄ =

tÜçäÉ=åìãÄÉêW=å==

Page 327: 1300 Math Formulas

CHAPTER 11. SERIES

317

1248. ( ) ( )∑∞

=

++=Nå

ååM åñëáåÄåñÅçë~

O

~ñÑ =

=

1249. ( )∫π

π−π= ÇñåñÅçëñÑ

N~å ==

=

1250. ( )∫π

π−π= ÇñåñëáåñÑ

NÄå ==

==

Page 328: 1300 Math Formulas

318

Chapter 12

Probability ====

12.1 Permutations and Combinations =

mÉêãìí~íáçåëW= ãå m =

`çãÄáå~íáçåëW= ãå ` =

tÜçäÉ=åìãÄÉêëW=åI=ã===

1251. c~Åíçêá~ä=( )( )åNåOåPON>å −−⋅⋅= K =

N>M = ==

1252. >åmåå = =

=

1253. ( )>ãå

>åmã

å

−= =

=1254. _áåçãá~ä=`çÉÑÑáÅáÉåí=

( )>ãå>ã

ã

å`ã

å

−=

= =

=1255. ãå

åã

å `` −= ==

1256. NãNå

Nãå

ãå ``` +

++ =+ =

=

Page 329: 1300 Math Formulas

CHAPTER 12. PROBABILITY

319

1257. åå

åO

åN

åM

å O```` =++++ K =

=1258. m~ëÅ~ä∞ë=qêá~åÖäÉ=

=oçï=M= = = = = = = N= = = = = = =oçï=N= = = = = = N= = N= = = = = =oçï=O= = = = = N= = O= = N= = = = =oçï=P= = = = N= = P= = P= = N= = = =oçï=Q= = = N= = Q= = S= = Q= = N= = =oçï=R= = N= = R= = NM= = NM= = R= = N= =oçï=S= N= = S= = NR= = OM= = NR= = S= = N=

===

12.2 Probability Formulas =bîÉåíëW=^I=_=mêçÄ~ÄáäáíóW=m=o~åÇçã=î~êá~ÄäÉëW=uI=vI=w=s~äìÉë=çÑ=ê~åÇçã=î~êá~ÄäÉëW=ñI=óI=ò=bñéÉÅíÉÇ=î~äìÉ=çÑ=uW=µ =^åó=éçëáíáîÉ=êÉ~ä=åìãÄÉêW= ε ==pí~åÇ~êÇ=ÇÉîá~íáçåW=σ =s~êá~åÅÉW= Oσ =aÉåëáíó=ÑìåÅíáçåëW= ( )ñÑ I= ( )íÑ ===

1259. mêçÄ~Äáäáíó=çÑ=~å=bîÉåí=

( )å

ã^m = I==

ïÜÉêÉ==ã=áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=éçëáíáîÉ=çìíÅçãÉëI==å=áë=íÜÉ=íçí~ä=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK==

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320

1260. o~åÖÉ=çÑ=mêçÄ~Äáäáíó=s~äìÉë=( ) N^mM ≤≤ =

=1261. ̀ Éêí~áå=bîÉåí=

( ) N^m = ==

1262. fãéçëëáÄäÉ=bîÉåí=( ) M^m = =

=1263. ̀ çãéäÉãÉåí=

( ) ( )^mN^m −= ==

1264. fåÇÉéÉåÇÉåí=bîÉåíë=( ) ( )^m_L^m = I==( ) ( )_m^L_m = =

=1265. ̂ ÇÇáíáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=

( ) ( ) ( )_m^m_^m +=∪ ==

1266. jìäíáéäáÅ~íáçå=oìäÉ=Ñçê=fåÇÉéÉåÇÉåí=bîÉåíë=( ) ( ) ( )_m^m_^m ⋅=∩ =

=1267. dÉåÉê~ä=^ÇÇáíáçå=oìäÉ=

( ) ( ) ( ) ( )_^m_m^m_^m ∩−+=∪ I==ïÜÉêÉ==

_^∪ =áë=íÜÉ=ìåáçå=çÑ=ÉîÉåíë=^=~åÇ=_I==_^∩ =áë=íÜÉ=áåíÉêëÉÅíáçå=çÑ=ÉîÉåíë=^=~åÇ=_K=

=1268. ̀ çåÇáíáçå~ä=mêçÄ~Äáäáíó=

( ) ( )( )_m

_^m_L^m

∩= =

=1269. ( ) ( ) ( ) ( ) ( )^L_m^m_L^m_m_^m ⋅=⋅=∩ =

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CHAPTER 12. PROBABILITY

321

1270. i~ï=çÑ=qçí~ä=mêçÄ~Äáäáíó=

( ) ( ) ( )∑=

Nááá _L^m_m^m I==

ïÜÉêÉ= á_ =áë=~=ëÉèìÉåÅÉ=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíëK===

1271. _~óÉë∞=qÜÉçêÉã=

( ) ( ) ( )( )^m

_m_L^m^L_m

⋅= =

=1272. _~óÉë∞=cçêãìä~=

( ) ( ) ( )

( ) ( )∑=

⋅=

ã

Nâáá

ááá

_L^m_m

_L^m_m^L_m I==

ïÜÉêÉ==

á_ =áë=~=ëÉí=çÑ=ãìíì~ääó=ÉñÅäìëáîÉ=ÉîÉåíë=EÜóéçíÜÉëÉëFI=^ =áë=íÜÉ=Ñáå~ä=ÉîÉåíI==( )á_m =~êÉ=íÜÉ=éêáçê=éêçÄ~ÄáäáíáÉëI=( )^L_m á =~êÉ=íÜÉ=éçëíÉêáçê=éêçÄ~ÄáäáíáÉëK=

=1273. i~ï=çÑ=i~êÖÉ=kìãÄÉêë=

pm å →

ε≥µ− =~ë= ∞→å I==

pm å →

ε<µ− =~ë= ∞→å I==

ïÜÉêÉ==

åp =áë=íÜÉ=ëìã=çÑ=ê~åÇçã=î~êá~ÄäÉëI=å =áë=íÜÉ=åìãÄÉê=çÑ=éçëëáÄäÉ=çìíÅçãÉëK==

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