Question

$\int \frac{\left(1+x\right)e^{x}}{\sin^{2} \left(xe^{x}\right)} dx $ is equal to

• A. $- \cot\, \left(e^{x}\right)+C$
• B. $\tan\,\left(xe^{x}\right)+C$
• C. $\tan \, \left(e^{x}\right)+C$
• D. $\cot \, \left(xe^{x}\right)+C$
• E. $- \cot \left(xe^{x}\right)+C$

Answer 1

Answer: (E)

Let $I=\int \frac{(1+x) e^{x}}{\sin ^{2}\left(x e^{x}\right)} d x$
Put $x e^{x}=t \Rightarrow \left(1 \cdot e^{x}+x \cdot e^{x}\right) \,d x=d t$
$\Rightarrow (1+x) e^{x} \,d x=d t$
$\therefore I=\int \frac{d t}{\sin ^{2} t}=\int cose c^{2}\, t\, d t$
$=-\cot \,t+C$
$=-\cot \left(x e^{x}\right)+C$