1300 Math Formulas - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas =...

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Page 1: 1300 Math Formulas - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë
Page 2: 1300 Math Formulas - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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=

Page 8: 1300 Math Formulas - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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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 ====

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CHAPTER 1. NUMBER SETS

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CHAPTER 1. NUMBER SETS

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CHAPTER 1. NUMBER SETS

6

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CHAPTER 1. NUMBER SETS

7

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CHAPTER 1. NUMBER SETS

8

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CHAPTER 1. NUMBER SETS

9

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CHAPTER 1. NUMBER SETS

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CHAPTER 1. NUMBER SETS

11

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12

Chapter 2

Algebra ====

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CHAPTER 2. ALGEBRA

13

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

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CHAPTER 2. ALGEBRA

14

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

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CHAPTER 2. ALGEBRA

15

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

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CHAPTER 2. ALGEBRA

16

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CHAPTER 2. ALGEBRA

17

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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 =

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CHAPTER 2. ALGEBRA

19

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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==

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CHAPTER 2. ALGEBRA

21

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CHAPTER 2. ALGEBRA

22

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CHAPTER 2. ALGEBRA

23

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24

Chapter 3

Geometry ====

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CHAPTER 3. GEOMETRY

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27

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CHAPTER 3. GEOMETRY

28

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CHAPTER 3. GEOMETRY

29

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CHAPTER 3. GEOMETRY

30

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CHAPTER 3. GEOMETRY

31

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CHAPTER 3. GEOMETRY

32

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33

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34

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CHAPTER 3. GEOMETRY

35

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CHAPTER 3. GEOMETRY

36

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38

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CHAPTER 3. GEOMETRY

40

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CHAPTER 3. GEOMETRY

43

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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=

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CHAPTER 3. GEOMETRY

44

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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===

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CHAPTER 3. GEOMETRY

46

245. ϕ== ëáåÇÇO

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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===

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CHAPTER 3. GEOMETRY

47

248. ϕ= ëáåÇÇO

Np ON =

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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===

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CHAPTER 3. GEOMETRY

48

251. ~o = ==

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253. O

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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===

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CHAPTER 3. GEOMETRY

49

==

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CHAPTER 3. GEOMETRY

50

ïÜÉêÉ=O

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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===

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CHAPTER 3. GEOMETRY

51

261. ONON ÄÄ~~ = ==

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CHAPTER 3. GEOMETRY

52

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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===

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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===

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CHAPTER 3. GEOMETRY

55

274. ( )[ ] ( )ñëáåñO

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CHAPTER 3. GEOMETRY

56

277. O

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

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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===

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CHAPTER 3. GEOMETRY

58

286. Üps _= ==

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

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CHAPTER 3. GEOMETRY

59

289. Q

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CHAPTER 3. GEOMETRY

60

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CHAPTER 3. GEOMETRY

61

3.27 Frustum of a Regular Pyramid =

_~ëÉ=~åÇ=íçé=ëáÇÉ=äÉåÖíÜëW=

åPON

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CHAPTER 3. GEOMETRY

62

299. O

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CHAPTER 3. GEOMETRY

63

==

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

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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==

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CHAPTER 3. GEOMETRY

65

311. P~Op O= ==

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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= =

=

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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==

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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= =

=

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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==

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CHAPTER 3. GEOMETRY

79

== =Picture 57.

=360. oêQp Oπ= ==361. OOoêOs π= =

==

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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 =

===

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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 =

=

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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=

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CHAPTER 4. TRIGONOMETRY

83

==

Figure 60. =

375. q~åÖÉåí=cìåÅíáçå=

ñí~åó = I= ( )O

NâOñπ

+≠ I= Kñí~å ∞≤≤∞− =

=

==

Figure 61. =

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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=

==

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CHAPTER 4. TRIGONOMETRY

85

==

Figure 63. =

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

=

Figure 64.

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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.

==========

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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 =

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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 =

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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 =

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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 =

======

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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 =

=

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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 =

=

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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 −= ====

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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

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CHAPTER 8. DIFFERENTIAL CALCULUS

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CHAPTER 8. DIFFERENTIAL CALCULUS

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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===

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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∆ ===

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222

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CHAPTER 8. DIFFERENTIAL CALCULUS

223

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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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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

∇−⋅∇∇=∇−= ===

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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 +=∫ =

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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=

=

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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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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

++−

=+∫ =

=

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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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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 +++++=+∫ =

=

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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 +−=−∫ =

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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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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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CHAPTER 9. INTEGRAL CALCULUS

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===

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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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CHAPTER 11. SERIES

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 - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

CHAPTER 11. SERIES

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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CHAPTER 11. SERIES

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==

=

Page 320: 1300 Math Formulas - WordPress.comkumarvishwash.files.wordpress.com/2018/01/...1300 Math Formulas = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = fp_k= =VVQVNMTTQN= = `çéóêáÖÜí=«=OMMQ=^KpîáêáåK=^ää=oáÖÜíë

CHAPTER 11. SERIES

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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CHAPTER 11. SERIES

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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CHAPTER 11. SERIES

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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CHAPTER 11. SERIES

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

=

=====

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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==

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CHAPTER 11. SERIES

317

1248. ( ) ( )∑∞

=

++=Nå

ååM åñëáåÄåñÅçë~

O

~ñÑ =

=

1249. ( )∫π

π−π= ÇñåñÅçëñÑ

N~å ==

=

1250. ( )∫π

π−π= ÇñåñëáåñÑ

NÄå ==

==

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