Evaluate: _0^ /2 2 x x ^ -1 ( x) dx. - Vidyadip

Question

Evaluate: $\int\limits_0^{\pi/2} 2 \sin x \cos x \tan^{-1 }(\sin x) dx.$

✓

Answer

$\text{I}=\int\limits_{0}^{\pi/2} 2 \sin x \cos x \tan-1 (\sin x) dx = \int\limits_0^1\tan^{-1}\text{t}\cdot\text{(2t) dt}$
where $\sin x = t =\Bigg[\tan^{-1} t\times\text{t}^{2}\Bigg]_0^1-\int\limits_0^{1}\frac{{t}^{2}}{\text{t}^{2}+1}\text{dt}$
$=\frac{\pi}{4}-\int_0^1\Bigg(1-\frac{1}{\text{t}^{1}+1}\Bigg)\text{dt}$
$=\frac{\pi}{4}-[\text{t - tan}^{-1}\text{t}]^1_0$
$=\frac{\pi}{4}-\Bigg[1-\frac{\pi}{4}\Bigg]$
$=\Bigg(\frac{\pi}{2}-1\Bigg)$.

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

Maximize $Z = 3x_1 + 4x_2, $ if possible,
Subject to the constraints
$\text{x}_1-\text{x}_2\leq-1$
$-\text{x}_1+\text{x}_2\leq0$
$\text{x}_1,\text{x}_2\geq0$

Find the angle between the following pairs of lines:$\frac{-\text{x}+2}{-2}=\frac{\text{y}-1}{7}=\frac{\text{z}+3}{-3}$ and $\frac{\text{x}+2}{-1}=\frac{2\text{y}-8}{4}=\frac{\text{z}-5}{4}$

Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in

Exactly one college.
At most two colleges.
At least two colleges.

Using the method of interation, find the area of the region bounded by the following lines:
3x - y - 3 = 0, 2x + y - 12 = 0, x - 2y - 1 = 0.

Find the distance of the point with position vector $-\hat{\text{i}}-5\hat{\text{j}}-10\hat{\text{k}}$ from the point of intersection of the line $\vec{\text{r}}=(2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})+\lambda(3\hat{\text{i}}+4\hat{\text{j}}+12\hat{\text{k}})$ with the plane $\vec{\text{r}}.(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}})=5.$

Find the shortest distance between the following pairs of lines whose vector equation are:
$\vec{\text{r}}=\big(\hat{\text{i}}+\hat{\text{j}}\big)+\lambda\big(2\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and $\vec{\text{r}}=2\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-5\hat{\text{j}}+2\hat{\text{k}}\big)$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{\text{x}}+\sqrt{\text{a}}}{1-\sqrt{\text{xa}}}\Big)$

Find the shortest distance between the lines $\frac{\text{x}+1}{7}=\frac{\text{y}+1}{-6}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-3}{1}=\frac{\text{y}-5}{-2}=\frac{\text{z}-7}{1}.$

Find the absolute maximum and the absolute minimum value of the following functions in the given intervals:
$f(x) = 3x^4 - 8x^3 + 12x^2 - 48x + 25$ in $[0,3]$

A bag contains 4 white, 7 black and 5 red balls. Three balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively.