Integrate the function x ^ -1 x - Vidyadip

Question

Integrate the function $x \tan^{-1}x$

✓

Answer

Let $I = \int {x{{\tan }^{ - 1}}x} dx$$ = \int {\left( {{{\tan }^{ - 1}}x} \right).x} dx$
$= \left( {{{\tan }^{ - 1}}x} \right).\frac{{{x^2}}}{2} - \int {\frac{1}{{1 + {x^2}}}.\frac{{{x^2}}}{2}dx} $
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\frac{{{x^2}}}{{1 + {x^2}}}dx} $
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\frac{{{x^2} + 1 - 1}}{{{x^2} + 1}}dx}$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\int {\left( {1 - \frac{1}{{{x^2} + 1}}} \right)dx}$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{1}{2}\left( {x - {{\tan }^{ - 1}}x} \right) + c$
$= \frac{1}{2}\left[ {{x^2}{{\tan }^{ - 1}}x - x + {{\tan }^{ - 1}}x} \right] + c$
$= \frac{{{x^2}}}{2}{\tan ^{ - 1}}x - \frac{x}{2} + \frac{1}{2}{\tan ^{ - 1}}x + c$

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 "1 Marks Question" 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

Show the relation R in the set A = {x $\in$ Z : 0 $\leq$ x $\leq$12}, given by R = {(a, b) : a = b}, is an equivalence relation. Find the set of all elements related to 1 in each case.

Given that the events A and B are such that P(A) = $\frac{1}{2}$, $P(A \cup B)=\frac{3}{5}$ and P(B) = p.
Find p if they are independent.

Find $\int e^{x} \sin x d x$

What positive value of x makes the following pair of determinants equal? .
$\begin{vmatrix}\text{2x}&3\\5&\text{x} \end{vmatrix}, \begin{vmatrix}\text{16}&3\\5&\text{2} \end{vmatrix}$

A fair coin is tossed 8 times, find the probability of.
exactly 5 heads.

Evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}$

A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at the rate of 0.05 cm/s. Find the rate at which its area is increasing when radius is 3.2 cm.

If $\vec{\text{a}}, \text{ }\vec{\text{b}},\text{ } \vec{\text{c}}$ are unit vectors such that $\vec{\text{a}}, \text{ }\vec{\text{b}}, \text{ }\vec{\text{c}}\text{ }=\vec{0},$ then write the value of $\vec{\text{a}} \text{ . }\vec{\text{b}} + \vec{\text{b}} \text{ . } \text{ }\vec{\text{c}} +\vec{\text{c}} . \vec{\text{a}}\text{ }.$

What is the cosine of the angle which the vector$\sqrt{2}\hat{\text{ i}}+\hat{\text{j}}+\hat{\text{k}}$ .

Let $\text{A} = \begin{bmatrix}2&4\\3&2\end{bmatrix}, \text{B} = \begin{bmatrix}1&3\\-2&5\end{bmatrix},\text{C} = \begin{bmatrix}-2&5\\3&4\end{bmatrix}.$Find each of the following:$\text{3A - C}$