Question

The value of $\int\limits_{0}^{\infty} \frac{dx}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$ is

• A. $\frac{\pi}{60}$
• B. $\frac{\pi}{20}$
• C. $\frac{\pi}{40}$
• D. $\frac{\pi}{80}$

Answer 1

Answer: (A)

Let $I =\int\limits_{0}^{\infty} \frac{dx}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$

$=\frac{1}{5}\left(\int\limits_{0}^{\infty} \frac{1}{\left(x^{2}+4\right)}dx-\int\limits_{0}^{\infty}\frac{1}{\left(x^{2}+9\right)}dx\right)$

$=\frac{1}{5} \left[\left[\frac{1}{2}tan^{-1}\frac{x}{2}\right]_{0}^{\infty}-\left[\frac{1}{3}tan^{-1}\frac{x}{3}\right]_{0}^{\infty}\right]$

$=\frac{1}{5}\left[\frac{1}{2}\cdot\frac{\pi}{2}-0-\left(\frac{1}{3}\cdot\frac{\pi}{2}-0\right)\right]$

$=\frac{1}{5} \left(\frac{\pi}{4}-\frac{\pi}{6}\right)$

$=\frac{\pi}{60}$