Evaluate: ^3 2 _ 0 |x x| dx - Vidyadip

Question

Evaluate: $\int\limits^\frac{3}{2}_{0}|x \cos\pi x| dx$

✓

Answer

$\int\limits^\frac{3}{2}_{0}|\text{x} \cos\pi \text{ x}| \text{dx} = \int\limits^{1/2}_{0}\text{x}\cos\pi \text{x dx}{-}\int\limits^{3/2}_{1/2}\text{x}\cos\pi \text{x dx}$
$= \left\{\frac{\text{x}\sin\pi \text{x}}{\pi} + \frac{\cos\pi\text{x}}{\pi^{2}}\right\}^{1/2}_{0}= \left\{\frac{\text{x}\sin\pi \text{x}}{\pi} + \frac{\cos\pi\text{x}}{\pi^{2}}\right\}^{3/2}_{1/2}$
$=\frac{1}{2\pi} - \frac{1}{\pi^{2}} - \bigg(-\frac{3}{2\pi}-\frac{1}{2\pi}\bigg) = \frac{5}{2\pi}-\frac{1}{\pi^{2}}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

A dice rolled two times and sum of appeared number found 7 . Find the conditional probability of getting 3 at least one time.

There are two bags, one of which contains $3$ black and $4$ white balls while the other contains $4$ black and $3$ white balls. $A$ die is thrown. If it shows up $1$ or $3,$ a ball is taken from the Ist bag; but it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a black ball.

Show that the derivative of the function $f $ given by $f(x) = 2x^3 - 9x^2 + 12x + 9,$ at $x = 1$ and $x = 2$ are equal.

Find the equation of the plane through the intersection of the planes 3x - 4y + 5z = 10 and 2x + 2y - 3z = 4 and parallel to the line x = 2y = 3z.

By Using properties of definite integrals, evaluate the following integral in Exercise:
$\int^{\pi}_{0}\log(1+\cos\text{x})\text{dx}$

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.

An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.

If $\text{y}=(\cot^{-1}\text{x})^2$ prove that $\text{y}^2(\text{x}^2+1)^2+2\text{x}(\text{x}^2+1)\text{y}_1=2.$

Find the dimensions of the rectangle of perimeter 36cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also find the maximum volume.

Find non-zero values of x satisfying the matrix equation:$\text{x}\begin{bmatrix}2\text{x}&2\\3&\text{x}\end{bmatrix}+2\begin{bmatrix}8&5\text{x}\\4&4\text{x}\end{bmatrix}=2\begin{bmatrix}(\text{x}^2+8)&24\$10)&6\text{x}\end{bmatrix}$