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 "4 Marks" 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

Find the general solution of the differential equation $\frac{\text{dy}}{\text{dx}}-{\text{y}}=\cos\text{x}$

At what points on the following curves, is the tangent parallel to x-axis?
y = 12(x + 1)(x - 2) on [-1, 2]

Evaluate the following intregals:
$\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$

Show that $\tan\Big(\frac{1}{2}\sin^{-1}\frac{3}{4}\Big)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?

Find a solution of $ x \left( x ^ { 2 } - 1 \right) \frac { d y } { d x } = 1 , $ which satisfy the condition y = 0 when x = 2.

Find the equation of a plane passing through the points (0, 0, 0) and (3, -1,2) and parallel to the line $\frac{\text{x}-4}{1}=\frac{\text{y}+3}{-4}=\frac{\text{z}+1}{7}$

using interation, find the area of the region bounded by the triangle ABC vertices A, B, C are (-1, 1), (0, 5) and (3, 2) respectivrly.

If x^x + y^x = 1, prove that $\frac{\text{dy}}{\text{dx}}=-\Big\{\frac{\text{x}^\text{x}(1+\log\text{x})+\text{y}^\text{x}\times\log\text{y}}{\text{x}\times\text{y}^{\text{x}-1}}\Big\}$

If $\text{A}=\begin{bmatrix}1&1\\0&1\end{bmatrix},$ prove that $\text{A}^\text{n}=\begin{bmatrix}1&\text{n}\\0&1\end{bmatrix}$ for all positive integers n.

If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:

$\vec{\text{a}},\ \vec{\text{b}},\ 3\vec{\text{a}}-2\vec{\text{b}}$