Definite Integration

Prove that:

    \[\int_{b}^{a} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)}\,\textrm{d}x=\frac{\pi}{4\left(a-b\right)}-\frac{1}{a^{2}-b^{2}}\left(a\tan^{-1}{\frac{b}{a}}+b\tan^{-1}{\frac{a}{b}}\right)\]

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Source: Porter, R.I. (1963). Further Elementary Analysis (2nd ed.). London: G. Bell & Sons. (Miscellaneous Examples, Question 27, p. 222)

Author: ascklee

Dr. Lee teaches at the Wee Kim Wee School of Communication and Information at the Nanyang Technological University in Singapore. He founded The Mathematics Digital Library in 2013.

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