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

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

Answers (1)

best_answer

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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