#### Please solve RD Sharma class 12 chapter 19 Definite Integrals exercise Multiple choice question 13 maths textbook solution

$\frac{\pi}{60}$

Given:

$\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} d x=\frac{\pi}{2(a+b)(b+c)(c+a)}$

Hint:

Using given condition find a,b,c.

Explanation:

Given that

$\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} d x=\frac{\pi}{2(a+b)(b+c)(c+a)} \ldots(i)$

We have to evaluate

$\int_{0}^{\infty} \frac{d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$

Let

$I=\int_{0}^{\infty} \frac{d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)}$

Multiply and divide by $x^{2}$

$I=\int_{0}^{\infty} \frac{x^{2} d x}{\left(x^{2}+4\right)\left(x^{2}+9\right)\left(x^{2}+0\right)}$

On comparing with equ (i)

We get

\begin{aligned} &a^{2}=4 ; b^{2}=9 ; c^{2}=0 \\\\ &a=2 ; b=3 ; c=0 \end{aligned}

To given

$\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} d x=\frac{\pi}{2(a+b)(b+c)(c+a)}$

$\int_{0}^{\infty} \frac{x^{2}}{\left(x^{2}+a^{2}\right)\left(x^{2}+b^{2}\right)\left(x^{2}+c^{2}\right)} d x=\frac{\pi}{2(2+3)(3+0)(0+2)}$

$=\frac{\pi}{2 \times 5 \times 3 \times 2}$

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