First slide

Methods of integration

Question

$\int \frac{x^{2} - 1}{(x^{4} + 3 x^{2} + 1) \tan^{- 1} (x + \frac{1}{x})} dx$ is equal to

Moderate

A

$\tan^{- 1} (x + \frac{1}{x}) + C$
B

$\cot^{- 1} (x + \frac{1}{x}) + C$
C

$\log (x + \frac{1}{x}) + C$
D

$\log [\tan^{- 1} (x + \frac{1}{x})] + C$

Solution

Let
$\begin{array}{l} I = \int \frac{x^{2} - 1}{(x^{4} + 3 x^{2} + 1) \tan^{- 1} (x + \frac{1}{x})} dx \\ = \int \frac{(1 - \frac{1}{x^{2}})}{(x^{2} + \frac{1}{x^{2}} + 3) \tan^{- 1} (x + \frac{1}{x})} dx \\ put, x + \frac{1}{x} = t \Rightarrow (1 - \frac{1}{x^{2}}) dx = dt \\ and {(x + \frac{1}{x})}^{2} = t^{2} \Rightarrow x^{2} + \frac{1}{x^{2}} = t^{2} - 2 \end{array}$
$\begin{aligned} ∴ I = \int \frac{1}{(t^{2} + 1) \tan^{- 1} t} dt = \log (\tan^{- 1} t) + C \\ = \log [\tan^{- 1} (x + \frac{1}{x})] + C \end{aligned}$

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