e^x(1+x) ^2 (x e^x ) d x is equal to - Vidyadip

MCQ

$\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$ is equal to

✓
$\tan \left(x e^x\right)+c$
B
$\cot \left(x e^x\right)+c$
C
$\cot \left(e^x\right)+c$
D
$\tan \left[e^x(1+x)\right]+c$

✓

Answer

Correct option: A.

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

$\text {Let } I=\int \frac{e^x(1+x)}{\cos ^2\left(x e^x\right)} d x$
$\text { Put } x e^x=t$
$\Rightarrow\left(x e^x+e^x\right) d x=d t$
$\Rightarrow e^x(x+1) d x=d t$
$\therefore I=\int \frac{d t}{\cos ^2 t}=\int \sec ^2 t d t=\tan t+c$
$=\tan \left(x e^x\right)+c$

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