Question

Evaluate:

`int (e^x (1 + x))/(sin^2 (xe^x))dx`

Answer 1

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

Put xe^x = t 

∴ `[x d/dx (e^x) + e^x d/dx (x)] dx = dt`

∴ (xe^x + e^x . 1)dx = dt

∴ e^x(1 + x)dx = dt

∴ I = `int1/sin^2t dt`

= `int "cosec"^2t  dt`

= −cot t + c

= −cot(x · e^x) + c