e^ x (1+x) ^2(xe^ x ) dx= - Vidyadip

MCQ

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

A
$2\log_\text{e}\cos(\text{xe}^{\text{x}})+\text{C}$
B
$\sec(\text{xe}^{\text{x}})+\text{C}$
✓
$\tan(\text{xe}^{\text{x}})+\text{C}$
D
$\tan(\text{x}+\text{e}^{\text{x}})+\text{C}$

✓

Answer

Correct option: C.

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

$\text{I}=\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^{2}(\text{xe}^{\text{x}})}\text{ dx}$Put $\text{xe}^{\text{x}}=\text{t}$
$\text{e}^{\text{x}}(1+\text{x})\text{dx}=\text{dt}$
$\text{I}=\int\frac{\text{dt}}{\cos^2\text{t}}$
$=\text{I}=\int\sec^2\text{t dt}$
$\text{t}=\tan\text{t}+\text{C}$
$\text{I}=\tan(\text{xe}^{\text{x}})+\text{C}$

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