Evaluate: `inte^(xsinx+cosx){(x^4cos^3x-xsinx+cosx)/(x^2cos^2x)}dx`

Explanation

We have,
`inte^((xsinx+cosx)){(x^(4)cos^(3)x-xsinx+cosx)/(x^(2)cos^(2)x)}dx`
`=inte^((xsinx+cosx)){x^(2)cosx-(xsinx-cosx)/((xcosx)^(2))}dx`
`=inte^((xsinx+cosx))(x^(2)cosx)dx-inte^((xsinx+cosx))(d)/(dx)((1)/(xcosx))`
`=intx{e^((xsinx+cosx))*(xcosx)}dx-inte^((xsinxcosx))(d)/(dx)((1)/(xcosx))`
Integrating it by parts, we get
`inte^((xsinx+cosx)){(x^(4)cos^(3)x-xsinx+cosx)/(x^(2)cos^(2)x)}dx`
`=intxe^(xinx+cosx)d(xsinx+cosx)-inte^((xsinx+cosx)*d((1)/(xcosx))`
`=xe^(xsinx+cosx)-int1*e^(xsinx+cosx)dx-{e^(xsinx+cosx)*(1)/(xcosx)-inte^(xsinx+cosx)*(xcosx)/(xcosx)dx}+K`
`=xe^(xsinx+cosx)-(1)/(xcosx)e^(xsinx+cosx)+K`
`=(x-(1)/(xcosx)e^(xsinx+cosx))+K`, where `K` is a constant of integration.