Question

Solve:-
$\int \frac{e^x(1+x)}{{\cos }^2\left(x{e}^x\right)}dx$

Answer 1

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

Let $t=x{e}^x\Rightarrow dt=x{e}^x+e^{x}dx=(x+1)e^{x}dx$

$=\int \frac{dt}{{\cos }^{2}t}$

$=\int {\sec }^{2}tdt$

$=\tan t+c$

$=\tan \left(x{e}^x\right)+c$