First slide

Methods of integration

Question

$\int \frac{x^{2} - 1}{(x^{4} + 3 x^{2} + 1) \tan^{- 1} (x + \frac{1}{x})} d x =$

Moderate

A

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

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

$\ln |\tan^{- 1} (x + \frac{1}{x})| + C$
D

$\frac{1}{2} \ln |(x + \frac{1}{x})| + C$

Solution

$\begin{aligned} \begin{array}{ll} I = \int \frac{x^{2} - 1}{(x^{4} + 3 x^{2} + 1) \tan^{- 1} (x + \frac{1}{x})} d x \\ = \int \frac{1 - \frac{1}{x^{2}}}{(x^{2} + \frac{1}{x^{2}} + 3) \tan^{- 1} (x + \frac{1}{x})} d x \\ Put x + \frac{1}{x} = t \Rightarrow (1 - \frac{1}{x^{2}}) d x = d t and x^{2} + \frac{1}{x^{2}} + 2 = t^{2} \\ I = \int \frac{d t}{(t^{2} + 1) \tan^{- 1} t} = \ln |\tan^{- 1} t| + c \\ = \ln |\tan^{- 1} (x + \frac{1}{x})| + c \end{array} \end{aligned}$

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