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

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Answer:

\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}

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