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

MCQ

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

A
$\text{e}^{\text{x}}\cot\text{x}+\text{C}$
✓
$-\text{e}^{\text{x}}\cot\text{x}+\text{C}$
C
$\text{e}^{\text{x}}\text{cosec x}+\text{C}$
D
$-\text{e}^{\text{x}}\text{cosec x}+\text{C}$

✓

Answer

Correct option: B.

$-\text{e}^{\text{x}}\cot\text{x}+\text{C}$

$\text{I}=\int\text{e}^{\text{x}}(1-\cot\text{x}+\cot^2\text{x})\text{dx}$

$\text{I}=\int\text{e}^\text{x}(1+\cot^2\text{x}-\cot\text{x})\text{dx}$

$\text{I}=\int\text{e}^{\text{x}}(\text{cosec}^2\text{x}-\cot\text{x})\text{dx}$

Here, $\text{f(x)}=-\cot\text{x}$

$\text{f}'(\text{x})=\text{cosec}^2\text{x}$

$\text{I}=-\text{e}^{\text{x}}\cot\text{x}+\text{C}$

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