Evaluate the following integrals:

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 $t=x e^{x}+1$

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

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