Question

$\displaystyle \int \frac{\left(1+x\right)e^{x}}{cot\left(xe^{x}\right)} dx$ is equal to

• A. $log \left|cos\left(xe^{x}\right)\right|+C$
• B. $log \left|cot\left(xe^{-x}\right)\right|+C$
• C. $log \left|sec\left(xe^{-x}\right)\right|+C$
• D. $log \left|cos\left(xe^{-x}\right)\right|+C$
• E. $log \left|sec\left(xe^{x}\right)\right|+C$

Answer 1

Answer: (E)

Let $I=\int \frac{(1+x) e^{x}}{\cot \left(x e^{x}\right)} d x$
Put $x e^{x}=t$
$\Rightarrow \left(x e^{x}+e^{x}\right) d x=d t$
$\Rightarrow (x+1) e^{x} d x=d t$
$\therefore I=\int \frac{d t}{\cot t}$
$=\int \tan\, t\, d t$
$=\log |\sec t|+C$
Again, put $t=x e^{x}$
$\Rightarrow I=\log \left|\sec \left(x e^{x}\right)\right|+C$