Question

Evaluate :
`int [e^x ( 1 + x )]/[ cos^2 (xe^x)] dx`

Answer 1

Let I = `int  [ e^x ( 1 + x )]/[ cos^2 (xe^x)] dx`

Put x.e^x = t
Diff.w.r.t..x
`therefore x.e^x + e^x. 1 = dt/dx`

`therefore e^x ( 1 + x )dx = dt`

`therefore I = int dt/cos^2 t = int sec^2 t dt`

`therefore tan t + c`

= tan ( x.e^x ) + e