Question

$\int_{0}^{\infty}{\frac{xdx}{(1 + x)(1 + x^{2})} =}$

• A. 0
• B. $\pi/2$
• C. $\pi/4$
• D. 1

Answer 1

Answer: (C)

$\pi/4$

Answer 2

$\int_{0}^{\infty}{\frac{xdx}{(1 + x)(1 + x^{2})} = \int_{0}^{\infty}{\frac{- \frac{1}{2}dx}{(1 + x)} + \int_{0}^{\infty}{\frac{\left( \frac{1}{2}x + \frac{1}{2} \right)}{1 + x^{2}}dx}}}$

$= \left\lbrack \frac{- 1}{2}\log(1 + x) \right\rbrack_{0}^{\infty} + \frac{1}{2} \times \frac{1}{2}\lbrack\log(1 + x^{2})\rbrack_{0}^{\infty} + \frac{1}{2}\lbrack\tan^{- 1}x\rbrack_{0}^{\infty}$

$= 0 + 0 + \frac{1}{2}\left\lbrack \frac{\pi}{2} - 0 \right\rbrack = \frac{\pi}{4}$.