SampleProblemsForAntiderivativesAndDefiniteIntegrals-9.pdf...

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Differential CalculusMTH 62-141Problems about Antiderivatives and Definite Integrals1. Antiderivatives or Indefinite Integrals: Find the most general antiderivative (indefiniteintegral) of the function. (Check your answer by differentiation)(a) Sum rule with basic forms.i.f(x) = 8x9-3x6+ 12x3. Answer:45x10-37x7+ 3x4+Cii.f(x) = (x+3x)(4 + 5x). Answer: 2x5/2+157x7/3+83x3/2+ 3x4/3+Ciii.f(x) =x(2 +x2)2. Answer:(x2+2)36+Civ.f(x) =x4+3xx2. Answer:x33-6x+Cv.f(x) = cosx-3ex+1x-1x2. Answer: sinx-3ex+ ln|x|+1x+Cvi.f(x) =2+x21+x2. Answer: tan-1x+x+Cvii.f(x) = sinx-(1-x2)-1/2+ 3 sec2x. Answer:-cosx+ cos-1x+ 3 tanx+Cviii.f(x) = 5ex-4 secxtanx+ sinhx. Answer: 5ex-4 secx+ coshx+Cix.f(x) =sinh(x/2) + cosh(x/2)sinh(x/2)-cosh(x/2). Answer:-ex+Cx.f(x) =cosxsin2x+sinxcos2x. Answer:-cscx+ secx+Cxi.f(x) =sin 2xsinx+cos 2xcos2x. Answer: 2 sinx+ 2x-tanx+Cxii.f(x) =sinx+ sinxtan2xsec2x. Answer:-cosx+Cxiii.f(x) =sinx+ sinxtan2xsec2x. Answer:-cosx+Cxiv.f(x) =x2-1x4-1. Answer: tan-1x+Cxv.f(x) = sin2(tan-1x). Answer:x-tan-1x+C(b) Product rule(Integration by parts)i.A.f(x) =xsinx. Answer:-xcosx+ sinx+CB.f(x) =xcosx. Answer:xsinx+ cosx+CC.f(x) =x2cosx. Answer: 2xcosx+x2sinx-2 sinx+CD.f(x) =xcoshx. Answer:12((x-1)ex-(x+ 1)e-x) +CE.f(x) =xex. Answer:xex-ex+CF.f(x) =xe-x. Answer:-xe-x-e-x+CG.f(x) =x2e3x. Answer: (x2/3-2x/9 + 2/27)e3x+C
H.f(x) =xlnx. Answer:x2lnx2-x24+CI.f(x) =x5lnx. Answer:x6lnx6-x636+CJ.f(x) =lnxx2. Answer:-lnxx-1x+CK.f(x) =lnxx. Answer: 2x(lnx-2) +CL.f(x) =xsecxtanx. Answer:xsecx-ln|secx+ tanx|+Cii.A.f(x) =x5ex. Answer: (x5-5x4+ 20x3-60x2+ 120x-120)ex+CB.f(x) =x4sinx. Answer:-x4cosx+ 4x3sinx+ 12x2cosx-24xsinx-24 cosx+CC.f(x) = (x5+3x2)e-2x. Answer: (-x52-5x44-5x32-21x24-21x4-218)e-2x+Ciii.A.f(x) = lnx. Answer:xlnx-x+CB.f(x) = tan-1x. Answer:xtan-1x-ln(x2+1)2+CC.f(x) = sin-1x. Answer:xsin-1x+1-x2+CD.f(x) = cos-1x. Answer:xcos-1x-1-x2+CE.f(x) = tan-1(1/x). Answer:xtan-1(1/x) +12ln(1 +x2) +Civ.A.f(x) =x3ex2. Answer:12ex2(x2-1) +CB.f(x) =x5e-x2. Answer:-12e-x2(x4+ 2x2+ 2) +CC.f(x) =x5sin(x2). Answer:12(-x4cos(x2) + 2 cos(x2) + 2x2sin(x2)) +Cv.A.f(x) =exsinx. Answer:ex2(sinx-cosx) +CB.f(x) =excosx. Answer:ex2(sinx+ cosx) +Cvi.A. Show thatZsinnxdx=-1nsinn-1xcosx+n-1nZsinn-2xdxB. Show thatZcosnxdx=1ncosn-1xsinx+n-1nZcosn-2xdxC. Show thatZtannxdx=1n-1tann-1x-Ztann-2xdx(n6= 1)D. Show thatZsec

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