Definite Integration

Prove that \displaystyle \frac{d}{dx}\left(\frac{x}{x^{2}+1}\right)=\frac{2}{(x^{2}+1)^{2}}-\frac{1}{x^{2}+1}.

Deduce the values of \displaystyle \int_{0}^{1} \frac{1}{(x^{2}+1)^{2}}\,\textrm{d}x.

Source: Porter, R.I. (1963). Further Elementary Analysis (2nd ed.). London: G. Bell & Sons. (Miscellaneous Examples, Question 19, 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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