Evaluate: `intx/((x^2-a^2)(x^2-b^2))\ dx`

Explanation

`I=int frac{x d x}{(x^{2}-a^{2})(x^{2}-b^{2})}`

`text { Putting } x^{2}=t`

`Rightarrow 2 x d x=d t`

`Rightarrow x d x=frac{d t}{2}`

`therefore I=frac{1}{2} int frac{d t}{(t-a^{2})(t-b^{2})}`

`text { Let } frac{1}{(t-a^{2})(t-b^{2})}=frac{A}{t-a^{2}}+frac{B}{t-b^{2}}`

`Rightarrow frac{1}{(t-a^{2})(t-b^{2})}=frac{A(t-b^{2})+B(t-a^{2})}{(t-a^{2})(t-b^{2})}`

`Rightarrow 1=A(t-b^{2})+B(t-a^{2})`

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