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

Question

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

✓

Answer

We have,
$\text{I}=\int\limits^{\infty}_0\frac{\log\text{x}}{1+\text{x}^2}\text{ dx}$
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,
$\therefore\ \text{I}=\int\limits^{\frac{\pi}{2}}_0\frac{\log(\tan\theta)}{1+\tan^2\theta}\sec^2\theta\text{ d}\theta$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\log(\tan\theta)\text{d}\theta\ ...(\text{i})$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\log\Big[\tan\Big(\frac{\pi}{2}-\theta\Big)\Big]\text{d}\theta$ $\Bigg[\because\ \int\limits^{\text{a}}_{0}\text{f(x)}\text{dx}=\int\limits^{\text{a}}_{\text{0}}\text{f}(\text{a}-\text{x})\text{dx}\Bigg]$
$\Rightarrow\text{I}=\int\limits^{\frac{\pi}{2}}_0\log(\cot\theta)\text{d}\theta\ ....(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\frac{\pi}{2}}_0\log(\tan\theta)\text{d}\theta+\int\limits^{\frac{\pi}{2}}_0\log(\cot\theta)\text{d}\theta$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0\big[\log(\tan\theta)+\log(\cot\theta)\big]\text{d}\theta$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0\big[\log(\tan\theta\times\cot\theta)\big]\text{d}\theta$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0\big(\log1\big)\text{d}\theta$
$\Rightarrow2\text{I}=\int\limits^{\frac{\pi}{2}}_0(0)\text{d}\theta$
$\Rightarrow2\text{I}=0$
$\Rightarrow\text{I}=0$
$\therefore\ \int\limits^{\infty}_0\frac{\log\text{x}}{1+\text{x}^2}\text{ dx}=0$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find a point on the parabola $y = (x - 4)^2,$ where the tangent is parallel to the chord joining $(4, 0)$ and $(5, 1).$

A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = The number of heads is odd,
B = The number of tails is odd.

Differentiate the following functions with respect to x:
$\frac{2^\text{x}\cos\text{x}}{(\text{x}^2+3)^2}$

Without expanding, show that the values of the following determinant are zero: $\begin{vmatrix}1^2&2^2&3^2&4^2\\2^2&3^2&4^2&5^2\\3^2&4^2&5^2&6^2\\4^2&5^2&6^2&7^2\end{vmatrix}$

For the following matrices verify the distributivity of matrix, multiplication over matrix addtion i.e., A(B + C) = AB + AC.
$\text{A}=\begin{bmatrix}2&-1\\1&1\\-1&2\end{bmatrix},\text{B}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{C}=\begin{bmatrix}1&-1\\0&1\end{bmatrix}$

Evaluate the following integrals:
$\int\text{x}\cos^3\text{x}^2\sin\text{x}^2\text{ dx}$

A manufacturer has three machine I, II, III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.
The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table:

Item	Number of hours required on machines
	I	II	III
M	1	2	1
N	2	1	1.25

She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

If the points A(m, -1), B(2, 1) and C(4, 5) are collinear, find the value of m.

ABCD is aparallelogram. the position vectora of the points A, B and C are respectively, $4\hat{\text{i}}+5\hat{\text{j}}-10\hat{\text{k}},2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $-\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}.$ Find the vector equation of the line BD. Also, reduce it to cartesian form.

Show that area of the parallelogram whose diagonals ate given by $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{|\vec{\text{a}}\times\vec{\text{b}}|}{2}.$ Also, find the area of the parallelogram, whose diagonals are $2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}.$