∫x2−1xx4+3x2+1dx is equal to

A
$\log_{e} |x + \frac{1}{x} + \sqrt{x^{2} + \frac{1}{x^{2}} + 3}| + C$
B
$\log_{e} |x - \frac{1}{x} + \sqrt{x^{2} + \frac{1}{x^{2}} - 3}| + C$
C
$\log_{e} |x + \sqrt{x^{2} + 3}| + C$
D
none of these

Solution:

We have,

$\begin{array}{l} I = \int \frac{x^{2} - 1}{x \sqrt{x^{4} + 3 x^{2} + 1}} d x = \int \frac{1 - \frac{1}{x^{2}}}{\sqrt{x^{2} + 3 + \frac{1}{x^{2}}}} d x \\ \Rightarrow I = \int \frac{1}{\sqrt{{(x + \frac{1}{x})}^{2} + 1^{2}}} d (x + \frac{1}{x}) \\ \Rightarrow I = \log |x + \frac{1}{x} + \sqrt{{(x + \frac{1}{x})}^{2} + 1}| \end{array}$

$\int \frac{x^{2} - 1}{x \sqrt{x^{4} + 3 x^{2} + 1}} d x$ is equal to

Solution:

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