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

Question

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

✓

Answer

Let $\text{I}=\int_{0}^\limits{1}\frac{\tan^{-1}\text{x}}{1+\text{x}^2}\text{ dx}$ Then,
Let $\tan^{-1}\text{x}=\text{t}$ Then, $\frac{1}{1+\text{x}^2}\text{ dx}=\text{dt}$
When $\text{x}=0,\text{t}=0$ and $\text{x}=1,\text{t}=\frac{\pi}{4}$
$\therefore\ \text{I}=\int_{0}^\limits{\frac{\pi}{4}}\text{t}\text{ dt}$
$\Rightarrow\text{I}=\Big[\frac{\text{t}^2}{2}\Big]^{\frac{\pi}{4}}_0$
$\Rightarrow\text{I}=\frac{\pi^2}{32}$

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 whether the function is differentiable at x = 1 and x = 2
$\text{f(x)}=\begin{cases}\text{x} & \text{x}\leq1\\2-\text{x} & 1\leq\text{x}\leq2\\-2+3\text{x}&\text{x}>2\end{cases}$

If $\text{y}=\sin^{-1}\big(6\text{x}\sqrt{1-9\text{x}^2}\big), -\frac{1}{3\sqrt{2}}<\text{x}<\frac{1}{3\sqrt{2}},$ then find $\frac{\text{dy}}{\text{dx}}.$

Evaluate the following integrals:$\int\frac{1}{\sqrt{7-6\text{x}-\text{x}^2}}\text{ dx}$

Express the vector $\vec{\text{a}}=5\hat{\text{i}}-2\hat{\text{j}}+5\hat{\text{k}}$ as the sum of two vectors such that one is parallel to the vector $\vec{\text{b}}=3\hat{\text{i}}+\hat{\text{k}}$ and other is perpendicular to $\vec{\text{b}}.$

ABCD are four points in a plane and Q is the point of intersection of the lines joining the mid-points of AB and CD, BC and AD. Show that $\overrightarrow{\text{PA}}+\overrightarrow{\text{PB}}+\overrightarrow{\text{PC}}+\overrightarrow{\text{PD}}=4\ \overrightarrow{\text{PQ}}$, where P ia any point.

Write a value of $\int\text{e}^{\log\sin\text{x}}\cos\text{x}\text{ dx}$

The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given the number triples in $5$ hrs, find how many bacteria will be present after $10$ hours. Also find the time necessary for the number of bacteria to be $10$ times the number of initial present.

Evaluate:$\int\frac{1 - x^{2}}{x(1 - 2x)}\text{dx}$.

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{6}}\frac{1}{1+\sqrt{\tan\text{x}}}\text{ dx}$

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}\cos(\text{x}-\text{y})=1$