Question:
Evaluate the following integrals:
$\int \frac{(x+1) e^{x}}{\sin ^{2}\left(x e^{x}\right)} d x$
Solution:
Assume $x e^{x}=t$
$d\left(x e^{x}\right)=d t$
$\left(e^{x}+x e^{x}\right) d x=d t$
$e^{x}(1+x) d x=d t$
Substituting $\mathrm{t}$ and $\mathrm{dt}$
$\Rightarrow \int \frac{d t}{\sin ^{2} t}$
$\Rightarrow \int \csc ^{2} t d t$
$\Rightarrow-\cot t+c$
But $t=x e^{x}+1$
$\Rightarrow-\cot \left(x e^{x}\right)+c$