$\int\frac{e^x(1+x)dx}{\cos^2(e^xx)}$ is equal to

$\tan(xe^x)+c$, where c is constant of integration.

The correct answer is Option (4) → $\tan(xe^x)+c$, where c is constant of integration.

Let $u = x e^{x}$.

$\displaystyle \frac{du}{dx}=e^{x}(1+x)$

$\Rightarrow du = e^{x}(1+x)\,dx$

The integral becomes:

$\displaystyle \int \frac{e^{x}(1+x)\,dx}{\cos^{2}(x e^{x})} = \int \sec^{2}(u)\,du$

$=\tan(u)+C$

$\tan(x e^{x}) + C$