Question #73167
1 Answer
(sinx -xcosx)/(xsinx+cosx)+C
Explanation:
=(-xsecx)/(xsin x+cos x)+int (sec x + xsecx tanx)/(xsinx+cosx) dx=−xsecxxsinx+cosx+∫secx+xsecxtanxxsinx+cosxdx
But
=
1/cos^2x\cancel (xsinx +cosx)/\cancel(xsinx+cosx)
=sec^2x
then
= (-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
=(x^2sin^2x+\cancel(xsinxcosx)-\cancel(xsinxcosx)+x^2cos^2x)/(xsinx+cosx)^2
=x^2/(xsinx+cosx)^2