# I asn i do a sine sino du cost x tant f 411 l etc i

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I +asn = I do a- sine Sino du = cost x tant f / 4=1,1 × Etc I tan =/ Ted " - i lnlsino-1-xtarctanx-lnlpi.FI Sint = OPI hyp XL t bZ= I ii. gx;aI ax ' ' a ' Ii :O . E- Ift × b=i# Trig sub l sin 't Jsino ' = fsmojyoafncsi.no#=fsincIstoI.coso----Ssino- to = cos -0 t I' A ' + C fasho ) = -Fxi-E(arcslhXT
12 ) ) c°S × DX uv Judy u sub , then integration by parts u= IT , du = X Z DX du - Tedx X - du = DX g coff x2 = f coff = f cosy . y ( usinu - fsinull ) ] -xtsinxt-cox cosultx ) W=U du cosy U Sth U t C cosy ) U= ¥ 1) I DX = I + any , geo , ,,d , d " " " Sinn using cosy = I tarixseczxseoxtanxdx Think about the = Ill Seo - x ) Sec - xsecxtanxdx integral of this to choose ~ u secy , du secxtanxdx trig identity = 14 n' Hu ' ) du 2) ferxdx y=r × =o ! " " " d " = ÷ " ' Tu ' + C = Tseix-÷secs ×€ dy : Tx ' " or DX 2. Tx dy = DX I e " zrxdu clue " le " ] Lfc " u dy wi u du e " 2 Cue " e " ] dw=a v - en 2rxer × _e" ÷ I = I 4) de U thx Drin du ¥dx 141¥ ) dy I # du or Su ' " Ju " - + c Iflnx) " HFI ax I III = 1,1¥ . In . " i x . E- xdx du = 2X DX = arcsinxttfltudx ) u 112+42 = arosinx -1 y ' ' ' { Jtsinzt = arosinx-CI-xry.cz#61/tsintcostdt { Luv Judy ] sinzt Sint cost = T y t dv= sinzt I [ It cos it J - I coszt ] du I v= tcoszt I [ It cost + tzlcosztdt ]u=2t Edu at { tcoszt + f) cosy dy ] du Zdt C- ttcosztt # Sinn ] -10 If tztcosttytsinzt ) -1C
lecture poly nominal Division when degree den E deg numerator 4X HOW TO FACTOR × , I IYi¥dx - x3 xz x -11 - * * . ,***i÷ xn+i ' III ! ! 's " " " " " grouping X - Factor out an X4-X3-Xi partial Fraction o y 3 - XZ X t l " X ' " X 3 X ' -13 x 4 × T Factor out " I " x3-x2- × _ = # + ¥izt¥ , = x - ( x t ) ( x 1) = Cx - 1) ( x t ) 44 I Integrate 4x=nCx¥H ) + B ( x -11 ) -1C ( x t ) ' ( X 1) ( x -11 ) ( x 1) = ( x 1) ' ( x -11 ) Ex - + x + I dx + 1¥94 I ,dx l l l I ¥1 DX , U = x I , du I DX X -_ I , 4( 1) = All 1) ( Itt ) t BC I -11 ) -1 ( ( I 1) 2 I du = Intuit = In Ix II 4=1312 ) 113=27 - 2 I ¥2 DX ; u = X I , dy = I DX x = l ; 4ft ) = Al I l ) ( I -11 ) t BC It 1) + ( C I l ) - 2du=2dX z ) Tut du 2/4 Z 4= ( ( 4) FI I = ¥ × =o ; 4 Co ) = A Co 1) ( o -11 ) t 2( Otl ) l ( o t ) - 3 I ¥1 ; u :X -11 , du =L o = A + z I /A I # du ; Intuit = lnlxtll.tl/2txt1n1x-il-F-#xt ! J deg denom 7 deg namer 2 × 2 y +4 = A ( X ' -14 ) t ( B Xtc ) ( x ) NO DIVISION 2 × 2-11+42 = Ay + 13 × +4 X O ; 210 ) 0+4 = A (0+4) + ( B ( O ) TC ) ( o ) X ( H2 -14 ) 42+4 i Fi w ie = I ( X - +4=0 ) 4 = A (4) /A=lI deg 2 , \ , 412-14=0 irreducible × =zi ; zfzi ) ' Zi -14 = 1 ( o ) + ( Brit C) Zi f # Axt f ,Y DX 214 i ' ) zi -14=4131-1 ) t 2 ( Ci ) In 1 × 1 + I ¥ydXt f ¥4 DX C 8+4 ) zi = 4 Btzci 1 u XL -14 , dy = 2x , ÷du=dX Iet -2yd = Lgbt 20£ * Must equal eachother ! I ) # du = Ith ly I = Ith 1 × 2+41 4=-413 ; 2i=2Ci z I ¥ , ax ; ¥4 : I = It di Idu tax th ; FT g - Farc tan ( ¥ ) uz= , u= ¥ , du Edy lnlxlttzlnlx441-tzarctahlf.lt#
lecture 7.8 Improper Integrals f ! It dx us I ? Tax lnlxlb , = Ix Ib , = I f t ) = In b - Int = z I = In b As b a 17 IT dx final thot I ? Tax Lima into = a convergent , finite limit for area Divergent , infinite area improper Integrals require limits in one or both bounds DNE Ex ) IF cosxdx = sin XII = limsinb shmo b IA O Divergent ex ) / F e - " dx = f I e " du = Ie " = Ee 2 × 19 = lim I e 2b f Leo ) b -79 U = 2X , du = -2 DX = O + I convergent ex ) Ita e " dx = ex t ' a = e ' laing . = e - o = e convergent Two infinite bounds Note / Fa f Cx ) dx is only convergent if l E f- Cx ) and [ a FCX ) dx are both convergent for all c i. e cannot cancel divergent limits × Ex I % ¥49 DX x 7 tano T = tan a DX = 7 sea f de arctah ( ¥ ) = 0=1 ai .