Explanation

(a)
Let $I = \int_{0}^{\infty}\frac{dx}{\left( x^{2} + 4 \right)(x^{2} + 9)}$
$= \frac{1}{5}\left( \int_{0}^{\infty}\ \frac{1}{(x^{2} + 4)}dx - \int_{0}^{\infty}\frac{1}{\left( x^{2} + 9 \right)}dx \right)$
$= \frac{1}{5}\left\lbrack \left\lbrack \frac{1}{2}\tan^{- 1}\frac{x}{2} \right\rbrack_{0}^{\infty} - \left\lbrack \frac{1}{3}\tan^{- 1}\frac{x}{3} \right\rbrack_{0}^{\infty} \right\rbrack$
$= \frac{1}{5}\left\lbrack \frac{1}{2}.\frac{\pi}{2} - 0 - \left( \frac{1}{3}.\frac{\pi}{2} - 0 \right) \right\rbrack$
$= \frac{1}{5}\left( \frac{\pi}{4} - \frac{\pi}{6} \right) = \frac{\pi}{60}$