$\int e^{x} [\frac{1}{\sqrt{1 + x^{2}}} + \frac{1 - 2 x^{2}}{\sqrt{{(1 + x^{2})}^{5}}}] dx$

Moderate

A

$e^{x} [\frac{1}{\sqrt{1 + x^{2}}} + \frac{x}{\sqrt{{(1 + x^{2})}^{3}}}] + C$
B

$e^{x} [\frac{1}{\sqrt{1 + x^{2}}} - \frac{x}{\sqrt{{(1 + x^{2})}^{3}}}] + C$
C

$e^{x} [\frac{1}{\sqrt{1 + x^{2}}} + \frac{x}{\sqrt{{(1 + x^{2})}^{5}}}] + C$
D

$e^{x} [\frac{1}{\sqrt{1 - x^{2}}} - \frac{x}{\sqrt{(1 + x^{2})}}] + C$

Solution

$\begin{array}{l} I = \int e^{x} [\frac{1}{\sqrt{1 + x^{2}}} - \frac{x}{\sqrt{{(1 + x^{2})}^{3}}} + \frac{x}{\sqrt{{(1 + x^{2})}^{3}}} + \frac{1 - 2 x^{2}}{\sqrt{{(1 + x^{2})}^{5}}}] d x \\ I = e^{x} \frac{1}{\sqrt{1 + x^{2}}} + e^{x} \frac{x}{\sqrt{{(1 + x^{2})}^{3}}} + C \\ I = e^{x} [\frac{1}{\sqrt{1 + x^{2}}} + \frac{x}{\sqrt{{(1 + x^{2})}^{3}}}] + C \end{array}$
$[∵ using \int e^{x} [f (x) + f^{'} (x)] dx = e^{x} f (x) + C]$

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