The integral ∫xxsinx+cosx2dx is equal to (where C is a constant of integration) :

∫x2(xsinx+cosx)2dx  Multiply and divide by cosx  =∫xcosx·xcosx(xsinx+cosx)2dx Integration by parts=xcosx∫xcosx(xsinx+cosx)2dx-∫ddxxcosx-∫xcosx(xsinx+cosx)2dxdx                                 put xsinx+cosx=t                                ⇒(xcosx+sinx-sinx)dx=dt                                  ⇒ xcosx dx=dt=xcosx·-1(xsinx+cosx)+∫cosx+xsinxcos2x·1(xsinx+cosx)dx=-xcosx(xsinx+cosx)+tanx+C=tanx-xsecxxsinx+cosx+C