Evaluate the following integrals: ^ _0 x (1+x)(1+x^2) dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$

✓

Answer

We have
$\text{I}=\int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}$
Putting $\text{x}=\tan\theta$
$\text{dx}=\sec^2\theta\text{ d}\theta$
When $\text{x}\rightarrow0;\theta\rightarrow0$
and $\text{x}\rightarrow\infty;\theta\rightarrow\frac{\pi}{2}$
Now, integral becomes,
$\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan\theta}{(1+\tan\theta)\sec^2\theta }\sec^2\theta\text{ d}\theta$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\tan\theta}{1+\tan\theta}\text{ d}\theta$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\frac{\sin\theta}{\cos\theta}}{1+\frac{\sin\theta}{\cos\theta}}\text{ d}\theta$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\sin\theta}{\sin\theta+\cos\theta}\text{ d}\theta\ ....(\text{i})$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\sin\big(\frac{\pi}{2}-\theta\big)}{\sin\big(\frac{\pi}{2}-\theta\big)+\cos\big(\frac{\pi}{2}-\theta\big)}\text{ d}\theta$ $\Bigg[\because\ \int\limits^{\text{a}}_{0}\text{f(x)}\text{dx}=\int\limits^{\text{a}}_{\text{0}}\text{f}(\text{a}-\text{x})\text{dx}\Bigg]$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\cos\theta}{\cos\theta+\sin\theta}\text{ d}\theta$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\cos\theta}{\sin\theta+\cos\theta}\text{ d}\theta\ ....(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\sin\theta+\cos\theta}{\sin\theta+\cos\theta}\text{ d}\theta$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0\text{d}\theta$
$\Rightarrow2\text{I}=\frac{\pi}{2}$
$\Rightarrow\text{I}=\frac{\pi}{4}$
$\therefore\ \int\limits^{\infty}_0\frac{\text{x}}{(1+\text{x})(1+\text{x}^2)}\text{ dx}=\frac{\pi}{4}$

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 "5 Marks Questions" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following:
$\int\frac{\sin^6\text{x}+\cos^6\text{x}}{\sin^2\text{x}\cos^2\text{x}}\text{dx}$

If sin y = x sin (a + y), prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^{2}\text{(a + y)}}{\sin\text{a}}.$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{bx}}{\text{b}-\text{ax}}\Big)$

A factory uses three different resources for the manufacture of two different products, 20 units of the resources A, 12 units of B and 16 units of C being available. 1 unit of the first product requires 2, 2 and 4 units of the respective resources and 1 unit of the second product requires 4, 2 and 0 units of respective resources. It is known that the first product gives a profit of 2 monetary units per unit and the second 3. Formulate the linear programming problem. How many units of each product should be manufactured for maximizing the profit? Solve it graphically.

Show that the lines $\frac{\text{x}+1}{-3}=\frac{\text{y}-3}{2}=\frac{\text{z}+2}{1}$ and $\frac{\text{x}}{1}=\frac{\text{y}-7}{-3}=\frac{\text{z}+7}{2}$ are coplanar. Also, find the equation of the plane containing them.

$\text{If}\cos^{-1}\frac{x}{\text{a}} + \cos^{-1}\frac{y}{\text{b}} = \alpha, \text{Prove that}\frac{{x}^{2}}{\text{a}^{2}} - 2\frac{xy}{\text{ab}}\cos\alpha +\frac{{y}^{2}}{\text{b}^{2}} = \sin^{2}\alpha$

Show that the following system of linear equation is inconsistent:

Show that the matrix $\text{A}=\begin{bmatrix}5&3\\12&7\end{bmatrix}$ is root of the equation $A^2 - 12A - I = 0$.

If $\text{x}=\text{e}^{\cos2\text{t}}$ and $\text{y}=\text{e}^{\sin2\text{t}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$

Using the method of integration find the area of the region bounded by lines:
2x + y = 4, 3x - 2y = 6 and x - 3y + 5 = 0.