Given that `int_0^oo(x^2)/((x^2+a^2)(x^2+b^2)(x^2+c^2))dx=pi/(2(a+b)(b+c)(c+a))` the value of `int_0^oo(dx)/((x ^2+4)(x^2+9)),` is `pi/(60)` b. `pi/(40)` c. `pi/(20)` d. `pi/(80)`

Explanation

`frac{pi}{60} `

`int_{0}^{infty} frac{1}{(x^{2}+4)(x^{2}+9)} d x `

`=frac{1}{5} int_{0}^{infty} frac{1}{(x^{2}+4)}-frac{1}{(x^{2}+9)} d x `

`=frac{1}{5}[frac{1}{2} tan ^{-1} frac{x}{2}-frac{1}{3} tan ^{-1} frac{x}{3}]_{0}^{infty} `

`=frac{1}{5}[frac{1}{2} times frac{pi}{2}-frac{1}{3} times frac{pi}{2}] `

`=frac{1}{5} times frac{pi}{12} `

`=frac{pi}{60}`