∫dx(x^2+a^2)(x^2+b^2) =1a^2 b^2∫a^2 b^2(x^2+a^2)(x^2+b^2) =1a^2 b^2∫(x^2+a^2) (x^2+b^2)(x^2+a^2)(x^2+b^2)dx =1a^2 b^2∫dxx^2+b^2 1a^2 b^2∫dxx ...

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Integration of Piecewise Continuous Functions

∫dxx2 + a2x2 ...

Question

$\int \frac{d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

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Solution

$\int \frac{d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$
$= \frac{1}{a^{2} - b^{2}} \int \frac{a^{2} - b^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})}$
$= \frac{1}{a^{2} - b^{2}} \int \frac{(x^{2} + a^{2}) - (x^{2} + b^{2})}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x$
$= \frac{1}{a^{2} - b^{2}} \int \frac{d x}{x^{2} + b^{2}} - \frac{1}{a^{2} - b^{2}} \int \frac{d x}{x^{2} + a^{2}}$
$= \frac{1}{a^{2} - b^{2}} [\frac{1}{b} {tan}^{- 1} (\frac{x}{b}) - \frac{1}{a} {tan}^{- 1} (\frac{x}{a})] + c$

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