Question

please do it fast it's urgent....

Answer 1

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

$\int \frac{xe^{x}+e^{x}}{\cos^{2}(xe^{x})}dx$

Substitute
$x {e}^{x} = u$

$(xe^{x}+e^{x})dx=du$

$dx= \frac{du}{xe^{x}+e^{x}}$

$\int \frac{xe^{x}+e^{x}}{\cos^{2}(u)(xe^{x}+e^{x})}du$

$\int \frac{1}{\cos^{2}(u)}du$

$\int \sec^{2}(u) du$

$\tan(u) + C$

Undo the substitution,

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

= $\ \boxed { \tan(xe^{x}) + C }$