Question

Given that $ \int\limits_{0}^{\infty}\frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} dx$
= $ \frac{\pi}{2\left(a+b\right)\left(b+c\right)\left(c+a\right)'} $ then
$ \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
$I =\int\limits_{0}^{\infty} \frac{d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$
$=\int\limits_{0}^{\infty} \frac{x^{2}}{x^{2}\left(x^{2}+4\right)\left(x^{2}+9\right)} d x$
$=\int\limits_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+0\right)\left(x^{2}+4\right)\left(x^{2}+9\right)} d x$
$=\frac{\pi}{2(0+2)(2+3)(3+0)} $
$ {\left[\because \int\limits_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} d x\right.}\left.=\frac{\pi}{2(a+b)(b+c)(c+a)}\right]$
$=\frac{\pi}{60}$