The partial fractions of x2(x2+a2)(x2+b2) are

given fraction is x2(x2+a2)(x2+b2)let x2=at(t+a2)(t+b2)=1(a2−b2)[a2t−b2t(t+a2)(t+b2)]=1a2−b2[a2t+a2−b2t+b2]

The partial fractions of x2(x2+a2)(x2+b2) are

given fraction is x2(x2+a2)(x2+b2)let x2=at(t+a2)(t+b2)=1(a2−b2)[a2t−b2t(t+a2)(t+b2)]=1a2−b2[a2t+a2−b2t+b2]

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The partial fractions of $\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})}$ are

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$\frac{1}{a^{2} + b^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$

$\frac{1}{a^{2} - b^{2}} [\frac{1}{x^{2} + a^{2}} - \frac{1}{x^{2} + b^{2}}]$

$\frac{1}{b^{2} - a^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$

$\frac{1}{a^{2} - b^{2}} [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}]$

answer is C.

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Detailed Solution

given fraction is $\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

let $x^{2} = a$

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

$[\frac{a^{2} t - b^{2} t}{(t + a^{2}) (t + b^{2})}] = \frac{1}{a^{2} - b^{2}} [\frac{a^{2}}{t + a^{2}} - \frac{b^{2}}{t + b^{2}}]$

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The partial fractions of x2(x2+a2)(x2+b2) are

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The partial fractions of $\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})}$ are