Integration by subtitution :

Question

I am having problem in solving the integration:
$$\int \frac{e^x (x+1)}{\cos^2 (x e^x)} dx.$$ Please help

Answer 1

Let $xe^x=u$ and and $\mathrm du=(xe^x+e^x) \mathrm dx=e^x(x+1) \mathrm dx$ (the derivative follows from the product rule). This gives

$$\int \sec^2 u \mathrm du=\tan u+C=\tan (xe^x)+C.$$

EDIT: Fixed to account for edit in question.

Answer 2

Notice, $(\cos(xe^x))^2=\cos^2(xe^x)$, so one should get $$\int \frac{e^x(x+1)}{(\cos(xe^x))^2}\ dx$$ $$=\int \frac{d(xe^x)}{\cos^2(xe^x)}$$ $$=\int \sec^2(xe^x)d(xe^x)$$ $$=\color{red}{\tan(xe^x)+C}$$