Let's do some calculus! (24)

Calculus Level 3

$\large \displaystyle \int_{0}^{\infty} {\dfrac{x^{2}}{( x^{2} + a^{2} )( x^{2} + b^{2} )( x^{2} + c^{2} )}} \,dx = \dfrac{\pi}{2 ( a+b )( b+c )( c+a )}$

Given that the equation above holds true for constant reals $a$ , $b$ and $c$ , find the value of $\displaystyle \int_{0}^{\infty} {\dfrac{1}{( x^{2} + 4 ) ( x^{2} + 9 )}} \,dx$

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\dfrac{\pi}{80}

\dfrac{\pi}{10}

\dfrac{\pi}{40}

\dfrac{\pi}{90}

\dfrac{\pi}{60}

\dfrac{\pi}{30}

\dfrac{\pi}{20}

\dfrac{\pi}{50}

1 solution

Chew-Seong Cheong
Sep 24, 2016

Putting $a=0$ , $b=2$ and $c=3$ , then we have:

$\begin{aligned} \int_0^\infty \frac {x^2}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx & = \frac \pi {2(a+b)(b+c)(c+a)} \\ \int_0^\infty \frac {x^2}{(x^2+0^2)(x^2+2^2)(x^2+3^2)}dx & = \frac \pi {2(0+2)(2+3)(3+0)} \\ \int_0^\infty \frac 1{(x^2+4)(x^2+9^2)}dx & = \boxed{\dfrac \pi {60}} \end{aligned}$

Let's do some calculus! (24)

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