Evaluate the following integrals: ^ 3 2 _0 |x |dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\cos\pi\text{x}\big|\text{dx}$

✓

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

Determine the values of a, b, c for which the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin\text{(a}+1)\text{x}+\sin\text{x}}{\text{x}}, &\text{for}\text{ x}<0,&\\\text{ c},&\text{for x}=0\\\frac{\sqrt{\text{x}+\text{bx}^2}-\sqrt{\text{x}}}{\text{bx}^\frac{3}{2}},&\text{for x}>0\end{cases}$ is continuous at x = 0.

From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.

Show that the lines $\vec{\text{r}}=(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}})+\lambda(3\hat{\text{i}}-\hat{\text{j}})\ \text{and}\ \vec{\text{r}}=({4\hat{\text{i}}-\hat{\text{k}})+\mu(2\hat{\text{i}}+3\hat{\text{k}}})$intersect. Also find their point of intersection.

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\sin\text{x}+\cos\text{x}\text{ on }\Big[0,\frac{\pi}{2}\Big]$

Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).

Using direction ratios show that the points A(2, 3, -4) B(1, - 2, 3) and C(3, 8, -11) are collinear.

A wholesale dealer deals in two kinds, $A$ and $B ($say$)$ of mixture of nuts. Each $\ kg$ of mixture $A$ contains $60$ grams of almonds, $30$ grams of cashew nuts and $30$ grams of hazel nuts. Each $\ kg$ of mixture $B$ contains $30$ grams of almonds, $60$ grams of cashew nuts and $180$ grams of hazel nuts. The remainder of both mixtures is per nuts. The dealer is contemplating to use mixtures $A$ and $B$ to make a bag which will contain at least $240$ grams of almonds, $300$ grams of cashew nuts and $540$ grams of hazel nuts. Mixture A costs $Rs. 8$ per $\ kg.$ and mixture $B$ costs $Rs. 12$ per $\ kg$. Assuming that mixtures $A$ and $B$ are uniform, use graphical method to determine the number of $\ kg.$ of each mixture which he should use to minimise the cost of the bag.

find the area of the region included between the parabola $y^2 = x$ and the line $x + y = 2$.

Solve the following systems of homogeneous linear equations by matrix method: $x + y - z = 0 , x - 2y + z = 0a , 3x + 6y - 5z = 0$

Solve the following equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$