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

Question

Evaluate: $\int\limits^{3/2}_{0} |x \sin \pi \text{ } x| \text{dx}.$

✓

Answer

$\text{I} = \int\limits^{3/2}_{0} | \text{x} \sin \pi \text{ x}| \text{dx}$
$= \int\limits^{1}_{0} \text{x} \sin \pi \text{ x} . \text{dx} - \int\limits^{3/2}_{1} \text{x} \sin \pi \text{x dx}$
$= \bigg[-\text{x} \frac{\cos \pi \text{x}}{\pi} + \frac{\sin \pi\text{x}}{\pi^{2}} \bigg]^{1}_{0} - \bigg[-\frac{\text{x}\cos \pi \text{x}}{\pi} + \frac{\sin \pi \text{x}}{\pi}^{2} \bigg]^{3/2}_{1}$
$= \frac{2}{\pi} + \frac{1}{\pi^{2}}$

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 "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 integrals of the functions in Exercises:
$\sin^4\text{x}$

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?

Find $\frac{\text{dy}}{\text{dx}}$ when x and y are connected by the relation:
$\sin(\text{xy})+\frac{\text{x}}{\text{y}}=\text{x}^2-\text{y}$

Differentiate the following functions with respect to x:
$\text{x}^{(\sin\text{x}-\cos\text{x})}+\frac{\text{x}^2-1}{\text{x}^2+1}$

Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{1-\text{x}^2}\text{ on }[-1,1]$

Differentiate the following functions with respect to x:
$3\text{e}^{-3\text{x}}\log(1+\text{x})$

The relation S defined on the set R of all real number by the rule aSb iff a ≥ b is:

An equivalence relation.
Reflexive, transitive but not symmetric.
Symmetric, transitive but not reflexive.
Neither transitive nor reflexive but symmetric.

Evaluate the following integrals:
$\int\text{x}^3\tan^{-1}\text{x dx}$

Solve the differential equation $\cos ^2 x \frac{d y}{d x}+y=\tan x\left(0 \leq x \leq \frac{\pi}{2}\right)$

Find $\int\sec^3\text{xdx}.$