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 "3 Marks Question" 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

If the direction cosines of any variable line in its two adjacent position are $l , m , n$ and $l +\delta, m +\delta m$, $n +\delta n$ the angle between those position be $\delta \theta$, then prove that
$
(\delta \theta)^2=(\delta l)^2+(\delta m)^2+(\delta n)^2
$

Prove the following results:
$\tan^{-1}\frac{2}{3}=\frac{1}{2}\tan^{-1}\frac{12}{5}$

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z + 7 = 0. Also, find the distance between the two planes.

Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$Function $\text{y}=\text{e}^\text{-x}+2$

If the mean and variance of a binomial variate X are 2 and 1 respectively, find P (X > 1).

Find the vector equations of the following planes in scalar product form $(\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}):$
$\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+\lambda(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\mu(4\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}})$

Find gof and fog when $f : R \rightarrow R$ and $g : R \rightarrow R$ are defined by:
$f(x) = x^2 + 8$ and $g(x) = 3x^3 + 1$

Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that (i) the youngest is a girl, (ii) at least one is a girl?

Evaluate the following integrals:
$\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{1-\cos2\text{x}}\text{ dx}$

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$\text{e}^{\text{x}-\text{y}}=\log\Big(\frac{\text{x}}{\text{y}}\Big)$