Given that ^ _0 x^2 (x^2+a^2)(x^2+b^2)(x^2+c^2) dx= 2(a+b)(b+c)(c… - Vidyadip

Question

Given that $\int\limits^{\infty}_0\frac{\text{x}^2}{(\text{x}^2+\text{a}^2)(\text{x}^2+\text{b}^2)(\text{x}^2+\text{c}^2)}\text{ dx}=\frac{\pi}{2(\text{a}+\text{b})(\text{b}+\text{c})(\text{c}+\text{a})},$ the value of $\int\limits^\infty_0\frac{1}{(\text{x}^2+4)(\text{x}^2+9)},$ is:

$\frac{\pi}{60}$
$\frac{\pi}{20}$
$\frac{\pi}{40}$
$\frac{\pi}{80}$

✓

Answer

$\frac{\pi}{60}$

Solution:

$\int\limits^\infty_0\frac{1}{\big(\text{x}^2+4\big)\big(\text{x}^2+9\big)}\text{dx}$

$=\frac{1}{5}\int\limits^\infty_0\frac{1}{\big(\text{x}^2+4\big)}-\frac{1}{\big(\text{x}^2+9\big)}\text{dx}$

$=\frac{1}{5}\bigg[\frac{1}{2\tan^{-1}{}}\frac{\text{x}}{2}-\frac{1}{3}\tan^{-1}\frac{\text{x}}{3}\bigg]^\infty_0$

$=\frac{1}{5}\bigg[\frac{1}{2}\times\frac{\pi}{2}-\frac{1}{3}\times\frac{\pi}{2}\bigg]$

$=\frac{1}{5}\times\frac{\pi}{12}$

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

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The minimum value of Z = 3x + 5y subjected to constraints $\text{x}+3\text{y}\geq3,\text{x}+\text{y}\geq2,\text{x},\text{y}\geq0$ is:

5
7
10
11

Mark the correct alternative in the following question:

Which one is not a requirement of a binomial dstribution?

There are 2 outcomes for each trial.
There is a fixed number of trials.
The outcomes must be dependent on each other.
The probability of success must be the same for all the trials.

Find the value of $\sec^2(\tan^{-1}2)+\text{cosec}^2(\cot^{-1}3)$

12
5
15
9

Suppose the function $f (x) - f (2x)$ has the derivative $5$ at $x = 1$ and derivative $7$ at $x = 2$. The derivative of the function $f (x) - f (4x)$ at $x = 1$, has the value equal to

Let f(x) = |x| + |x - 1|, then:

f(x) is continuous at x = 0, as well as at x = 1
f(x) is continuous at x = 0, but not at x = 1
f(x) is continuous at x = 0, but not at x = 0
none of these

Let $f(x)=2 x^{2}-x-1$ and $S =\{n \in Z :|f(n)| \leq 800\}$ . Then value of $\sum_{n \in S} f(n)$ is . . . . .

Choose the correct answer from the given four options.
In a college, 30% students fail in physics, 25% fail in mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in physics if she has failed in mathematics is:

The function which is neither decreasing nor increasing in $\left( {{\pi \over 2},{{3\pi } \over 2}} \right)$ is

Let $f_n=\int \limits_0^{\frac{\pi}{2}}\left(\sum \limits_{k=1}^n \sin ^{k-1} x\right)\left(\sum \limits_{k=1}^n(2 k-1) \sin ^{k-1} x\right) \cos x$ $d x, n \in N$. Then $f_{21}-f_{20}$ is equal to $...........$.

The corner points of the feasible region determined by the system of linear constraints are $(0,0),(0,40),(20,40),(60,20),(60,0) .$ The objective function is $z=4 x+3 y$ Compare the quantity in Column $A$ and Column $B$

Column	Maximum of $z$
$A$	$300$
$B$	$325$