Evaluate the following integrals: _ ^ 3 2 1- 2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{1-\cos2\text{x}}\text{ dx}$

✓

Answer

$\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{1-\cos2\text{x}}\text{ dx}$
$=\int_{\pi}^\limits{\frac{3\pi}{2}}\sqrt{2\sin^2\text{x}}\text{ dx}$
$=\sqrt{2}\int_{\pi}^\limits{\frac{3\pi}{2}}\big[\sin\text{x}\big]\text{dx}$
$=\sqrt{2}\int_{\pi}^\limits{\frac{3\pi}{2}}\sin\text{x}\text{ dx}$ $(\sin\text{x}<0\text{ for }\pi\leq\text{x}\leq2\pi)$
$=\sqrt{2}\big[(-\cos\text{x})\big]^{\frac{3\pi}{2}}_\pi$
$=\sqrt{2}\Big(\cos\frac{3\pi}{2}-\cos\pi\Big)$
$=\sqrt{2}\big[0-(-1)\big]$
$=\sqrt{2}\times1$
$=\sqrt{2}$

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 "3 Marks Question" from Definite Integrals→Definite 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 box contains 13 bulbs, out of which 5 are defective. 3 bulbs are randomly drawn, one by one without replacement, from the box. Find the probability distribution of the number of defective bulbs.

Evaluate the following integrals:
$\int^\limits{\frac{\pi}{2}}_{0}|\cos2\text{x}|\text{dx}$

Let f be a function from R to R, such that f(x) = cos(x + 2). Is f invertible? Justify your answer.

Solve the following equation for x:
$3\sin^{-1}\frac{2\text{x}}{1+\text{x}^2}-4\cos^{-1}\frac{1-\text{x}^2}{1+\text{x}^2}+2\tan^{-1}\frac{2\text{x}}{1-\text{x}^2}=\frac{\pi}{3}$

Show that the line through the points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (-1, -2, 1) and, (1, 2, 5).

$\text{If y}=(\tan ^1 \text{x})^2,\text{show that }(\text{x}^2+1)^2 \ \text{y}_2+ 2\text{x}(\text{x}^2+1)_\text{y}=2$

Give an example of a function which is continuos but not differentiable at at a point.

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\overrightarrow{\text{OP}}=2\vec{\text{a}}+\vec{\text{b}}\text{ and }\overrightarrow{\text{OQ}}=\vec{\text{a}}-2\vec{\text{b}}$, respectively in the ratio 1 : 2 internally and externally.

Evaluate the following integrals:$\int\frac{1}{\sqrt{(1-\text{x}^2)\big\{9+\big(\sin^{-1}\text{x}\big)^2\big\}}}\text{ dx}.$

Find a unit vector perpendicular to both the vectors $4\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $-2\hat{\text{i}}+\hat{\text{j}}-2\hat{\text{k}.}$