Question

Topic :- Definite integral

please give step by step solution.

Answer 1

xcos(x)sin2(x)
∫xcos(x)sin2(x)dx∫xcos⁡(x)sin2⁡(x)dxIntegrate by parts: ∫fg′=fg−∫f′g∫fg′=fg−∫f′gff=x=x,g′g′=cos(x)sin2(x)=cos⁡(x)sin2⁡(x)↓↓ Steps↓↓ Stepsf′f′=1=1,gg=sin3(x)3=sin3⁡(x)3:=xsin3(x)3−∫sin3(x)3dx
∫sin3(x)3dx
=13∫sin3(x)dx
Now solving:∫sin3(x)dx∫sin3⁡(x)dxPrepare for substitution:=∫(1−cos2(x))sin(x)dx=∫(1−cos2⁡(x))sin⁡(x)dxSubstitute u=cos(x)u=cos⁡(x) ⟶⟶ dx=−1sin(x)dudx=−1sin⁡(x)du (Steps):=∫(u2−1)du=∫(u2−1)duApply linearity:=∫u2du−∫1du=∫u2du−∫1duNow solving:∫u2du∫u2duApply power rule:∫undu=un+1n+1∫undu=un+1n+1 with n=2n=2:=u33=u33Now solving:∫1du∫1duApply constant rule:=u=uPlug in solved integrals:∫u2du−∫1du∫u2du−∫1du=u33−u=u33−uUndo substitution u=cos(x)u=cos⁡(x):=cos3(x)3−cos(x)=cos3⁡(x)3−cos⁡(x)Plug in solved integrals:13∫sin3(x)dx13∫sin3⁡(x)dx=cos3(x)9−cos(x)3=cos3⁡(x)9−cos⁡(x)3Plug in solved integrals:xsin3(x)3−∫sin3(x)3dxxsin3⁡(x)3−∫sin3⁡(x)3dx=xsin3(x)3−cos3(x)9+cos(x)3=xsin3⁡(x)3−cos3⁡(x)9+cos⁡(x)3The problem is solved:∫xcos(x)sin2(x)dx∫xcos⁡(x)sin2⁡(x)dx=xsin3(x)3−cos3(x)9+cos(x)3+C=xsin3⁡(x)3−cos3⁡(x)9+cos⁡(x)3+CRewrite/simplify:=3xsin3(x)−cos3(x)+3cos(x)9+C
now substituting limits
−540sin(540)+cos(540)−1620sin(180)−9cos(180)36−29−540sin⁡(540)+cos⁡(540)−1620sin⁡(180)−9cos⁡(180)36−29Simplify:−540sin(540)−cos(540)+1620sin(180)+9cos(180)−836−540sin⁡(540)−cos⁡(540)+1620sin⁡(180)+9cos⁡(180)−836Approximation:−31.2508682721236

DO MARK BRAINLIEST
HOPE IT HELPS YOU