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

Question

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

✓

Answer

We have,
$\int_{0}^\limits{1}\text{x}\tan^{-1}\text{x}\text{ dx}=\tan^{-1}\text{x}\int_{0}^\limits{1}\text{x dx}-\int_{0}^\limits{1}\big(\int\text{dx}\big)\frac{\text{d}}{\text{dx}}\big(\tan^{-1}\text{x}\big)\text{dx}$
$=\Big[\frac{\text{x}^2}{2}\tan^{-1}\text{x}\Big]^1_0-\frac{1}{2}\int_{0}^\limits{1}\frac{\text{x}^2}{1+\text{x}^2}\text{ dx}$
$=\Big[\frac{\text{x}^2}{2}\tan^{-1}\text{x}\Big]^1_0-\frac{1}{2}\int_{0}^\limits{1}\frac{1+\text{x}^2-1}{1+\text{x}^2}\text{ dx}$
$=\frac{1}{2}\Big(\frac{\pi}{4}\Big)-\frac{1}{2}\Bigg[\int_{0}^\limits{1}\text{dx}-\int_{0}^\limits{1}\frac{\text{dx}}{1+\text{x}^2}\Bigg]$
$=\frac{\pi}{8}-\frac{1}{2}\big[\text{x}-\tan^{-1}\text{x}\big]^{1}_0$
$=\frac{\pi}{8}-\frac{1}{2}\Big[1-\frac{\pi}{4}\Big]$
$=\frac{\pi}{8}-\frac{1}{2}+\frac{\pi}{8}$
$=\frac{\pi}{4}-\frac{1}{2}$

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

Solve the following differential equations:$\text{cosec x}\log\text{y}\frac{\text{dy}}{\text{dx}}+\text{x}^2\text{y}^2=0$

If given function is continuous at $x=1$, then find $a$ and $b. f(x)=\left\{\begin{array}{cl} 3 a x+b, & \text { if } x>1 \\ 11, & \text { if } x=1 \\ 5 a x-2 b, & \text { if } x<1 \end{array}\right.$

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 $\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}}.$

Show that the differential equation of $\left\{x \cos \left(\frac{y}{x}\right)+y \sin \left(\frac{y}{x}\right)\right\} y d x=\left\{y \sin \left(\frac{y}{x}\right)-x \cos \left(\frac{y}{x}\right)\right\} x d y$ is homogeneous and solve it.

Differentiate the following functions with respect to x:
$(\sin\text{x})^{\cos\text{x}}$

Find a unit vector perpendicular to each of the vector $\vec a + \vec b\;$ and $\vec a - \vec b$, where $\vec a = 3\hat i + 2\hat j + 2\hat k\;$ and $\;\vec b = \hat i + 2\hat j - 2\hat k$.

An open box with square base is to be made of a given quantity of card board of area $c^2.$ Show that the maximum volume of the box is $\frac{\text{c}^2}{6\sqrt{3}}$ cubic units.

Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being seperated.

Show that the following system of linear equation is inconsistent:
$2x + 3y = 5$
$6x + 9y = 10$

Differentiate the following functions with respect to x:
$\log\sqrt{\frac{1-\cos\text{x}}{1+\cos\text{x}}}$