Question

Evaluate $\int \frac{e^x(1+x)}{{\sin }^2\left(x{e}^x\right)}dx$

Answer 1

$\int \frac{e^x(1+x)}{{\sin }^2\left(x{e}^x\right)}dx$

Let $x{e}^x=t$

$\therefore (e^x+x{e}^x)dx=dt$

$\therefore dx=\frac{dt}{e^x+x{e}^x}$

$\therefore \int \frac{e^x(1+x)}{{\sin }^{2}t}\times \frac{dt}{e^x(1+e^x)}$

$=\int cose{c}^{2}tdt$

$=-\cot t+c$

$=-\cot \left(x{e}^x\right)+c$.