Question:
Mark the correct alternative in each of the following:
Evaluate $\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)} d x=$
A. $2 \log _{e} \cos \left(x e^{x}\right)+C$
B. $\sec \left(x e^{x}\right)+C$
C. $\tan \left(x e^{x}\right)+C$
D. $\tan \left(x+e^{x}\right)+C$
Solution:
let $(\mathrm{t})=\mathrm{x} e^{x}$
$\frac{d t}{d x}=e^{x}(1+x)$
$\Rightarrow \int \frac{d t}{(\cos t)^{2}}=\int(\sec t)^{2} d t$
$=\tan t$
$\left(\right.$ put $\left.(\mathrm{t})=\mathrm{x} e^{x}\right)$
$=\tan \left(x e^{x}\right)+c$