∫x2−a2x2+a2 dx=∫(x2+a2)−2a2x2+a2=∫1−2a2x2+a2dx=x−2a2∫1x2+a2dx=x−2a21atan−1xa+c=x−2atan−1xa+c

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$\int \frac{x^{2} - a^{2}}{x^{2} + a^{2}} d x =$

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$x + 2 a \tan^{- 1} (x / a) + C$

$x - 2 a \tan^{- 1} (x / a) + C$

$2 x - a \tan^{- 1} (x / a) + C$

$x + 1 / 2 a \tan^{- 1} (x / a) + C$

answer is C.

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Detailed Solution

$\int \frac{x^{2} - a^{2}}{x^{2} + a^{2}} d x$

$= \int \frac{(x^{2} + a^{2}) - 2 a^{2}}{x^{2} + a^{2}}$

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

$= x - 2 a^{2} \int \frac{1}{x^{2} + a^{2}} d x$

$= x - 2 a^{2} (\frac{1}{a} \tan^{- 1} \frac{x}{a}) + c$

$= x - 2 a \tan^{- 1} \frac{x}{a} + c$

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