Question #73167

1 Answer
Dec 12, 2017

(sinx -xcosx)/(xsinx+cosx)+C

Explanation:

int (x^2 dx)/((x sin x +cos x)^2)= int -x/(cos x) d(1/(xsinx+cosx)) x2dx(xsinx+cosx)2=xcosxd(1xsinx+cosx)

=(-xsecx)/(xsin x+cos x)+int (sec x + xsecx tanx)/(xsinx+cosx) dx=xsecxxsinx+cosx+secx+xsecxtanxxsinx+cosxdx

But
(sec x + xsecx tanx)/(xsinx+cosx) =(1/(cos x) + (xsinx)/(cos^2x))/(xsin x+ cosx)secx+xsecxtanxxsinx+cosx=1cosx+xsinxcos2xxsinx+cosx

=1/cos^2x\cancel (xsinx +cosx)/\cancel(xsinx+cosx)
=sec^2x

then
int (x^2 dx)/((x sin x +cos x)^2) = (-xsecx)/(xsin x+cos x) + int sec^2x dx

= (-xsecx)/(xsin x+cos x)+tanx + C
= (xsin^2x + sin xcos x - x)/(cos x(xsin x+cos x))+C
=(sinx -xcosx)/(xsinx+cosx)+C

We can verify by differentiating

f'(x) = ((xsin x+cos x)(\cancel(cosx)-\cancel(cosx)+xsinx) - (sinx - xcosx)(xcosx + \cancel (sinx) - cancel(sinx)))/(xsinx+cosx)^2

=(x^2sin^2x+\cancel(xsinxcosx)-\cancel(xsinxcosx)+x^2cos^2x)/(xsinx+cosx)^2
= x^2/(xsinx+cosx)^2