x(x^2 a^2)(x^2 b^2) = x(x a)(x+a)(x b)(x+b) =a2a(a^2 b^2)(x a)+ a 2a(a^2 b^2)(x+a)+b2b(b^2 a^2)(x b)+ b 2b(b^2 a^2)(x+b) =12(a^2 b^2)( 1x a+ ...

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(a+b)(a-b) Expansion and Visualisation

xx2 - a2x2 -b...

Question

$\frac{x}{(x^{2} - a^{2}) (x^{2} - b^{2})}$

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Solution

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

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