Evaluate the following integrals: _ 0 ^1 ^ -1 ( 2x 1-x^2 )dx - Vidyadip

Question

Evaluate the following integrals:$\int_{0}^\limits{1}\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)\text{dx}$

✓

Answer

$\int_{0}^\limits{1}\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)\text{dx}$$=\int_{0}^\limits{1}2\tan^{-1}\text{x dx}$
$=2\Big[\text{x}\tan^{-1}\text{x}\Big]^1_0-2\int_{0}^\limits{1}\frac{\text{x}}{1+\text{x}^2}\text{ dx}$
$=2\Big[\text{x}\tan^{-1}\text{x}\Big]^1_0-\Big[\log\big(1+\log\text{x}^2\big)\Big]$
$=2\frac{\pi}{4}-0-\log2+0$
$=\frac{\pi}{2}-\log2$

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 "Solve the Following Question.(3 Marks)" from Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Obtain the differential equation by eliminating the arbitrary constants from the following equations:

$y=c_1 e^{2 x}+c_2 e^{5 x}$

Evalute the following integrals:
$\int\frac{\sec^2\text{x}}{\tan\text{x}+2}\text{dx}$

Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}\text{|x|}\cos\Big(\frac{1}{\text{x}}\Big), & \text{ x}\neq 0\\0 &\text{ x} = 0\end{cases}\text{at x}=0$

Check the commutativity and associativity of the following binary operations:
'*' on R defined by a * b = a + b - 7 for all a, b ∈ R.

Differentiate the following w. r. t. x.$\tan ^{-1}\left(\frac{5 x+1}{3-x-6 x^2}\right)$

Diffrentiate the following w. r. t. x.$\operatorname{cosec}^{-1}\left(\frac{1}{4 \cos ^3 2 x-3 \cos 2 x}\right)$

Solve the following equations by reduction method $:x + 3y = 2, 3x + 5y = 4.$

Find the principal values of the following:
$\tan^{-1}\Big(-\frac{1}{\sqrt3}\Big)$

Evaluate the following integrals:
$\int\sec\text{x}.\text{log} (\sec\text{x}+\tan\text{x})\text{dx}$

Write a value of $\int\text{e}^{3\log\text{x}}\text{x}^{4}\text{ dx}.$