Choose the correct answer in Exercises: e^ x (1+x) ^2(e^xx) dx eq… - Vidyadip

MCQ

Choose the correct answer in Exercises:
$\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^2(\text{e}^\text{x}\text{x})}\text{dx}$ equals

A
$-\cot(\text{e}^{\text{x}}\text{x})+\text{C}$
✓
$\tan(\text{xe}^\text{x})+\text{C}$
C
$\tan(\text{e}^\text{x})+\text{C}$
D
$\cot(\text{e}^\text{x})+\text{C}$

✓

Answer

Correct option: B.

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

$\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^2(\text{e}^\text{x}\text{x})}\text{dx}$
$\text{Let }\text{e}^{\text{x}}\text{x}=\text{t}$
$\Rightarrow(\text{e}^\text{x}\cdot\text{x}+\text{e}^{\text{x}}\cdot1)\text{dx}=\text{dt}$
$\text{e}^{\text{x}}(\text{x}+1)\text{dx}=\text{dt}$
$\therefore\int\frac{\text{e}^{\text{x}}(1+\text{x})}{\cos^2(\text{e}^\text{x}\text{x})}\text{dx}=\int\frac{\text{dt}}{\cos^2\text{t}}$
$=\int\sec^2\text{t}\text{ dt}$
$=\tan\text{t}+\text{C}$
$=\tan(\text{e}^{\text{x}}\cdot\text{x})+\text{C}$
Hence, the correct answer is B.

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