#160;Now, ∫x dx(x^2+a^2)(x^2+b^2) =12(b^2 a^2)[∫2x dx(x^2+a^2) . .∫2x dx(x^2+b^2)] =12(b^2 a^2)[∫ d(x^2+a^2)(x^2+a^2) . .∫d(x^2+b^2)( ...

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Functions

∫x dxx2+a2x2+...

Question

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

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Solution

Now,
$\int \frac{x d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$
$= \frac{1}{2 (b^{2} - a^{2})} [\int \frac{2 x d x}{(x^{2} + a^{2})} -$ $\int \frac{2 x d x}{(x^{2} + b^{2})}]$
$= \frac{1}{2 (b^{2} - a^{2})} [\int \frac{d (x^{2} + a^{2})}{(x^{2} + a^{2})} -$ $\int \frac{d (x^{2} + b^{2})}{(x^{2} + b^{2})}]$
$= \frac{1}{2 (b^{2} - a^{2})} [log (x^{2} + a^{2}) - log (x^{2} + b^{2})] + c$ [ Where $c$ being integrating constant]
$= \frac{1}{2 (b^{2} - a^{2})} [log \frac{(x^{2} + a^{2})}{(x^{2} + b^{2})}] + c$

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

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