^ 2 _ 3 1+ (1- )^ 5 2 - Vidyadip

Question

$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{5}{2}}}$

✓

Answer

Let $\text{I}=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^{\frac{5}{2}}}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{\sqrt{1+\cos\text{x}}}{(1-\cos\text{x})^2\sqrt{1-\cos\text{x}}}\text{dx}$
$\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(1-\cos\text{x})^2}\text{dx}=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\frac{1}{(\sin^2\text{x})}\text{dx}$
$=\int\limits^{\frac{\pi}{2}}_{\frac{\pi}{3}}\cos\text{ec}^2\text{x dx}=\big[-\cot\text{x}\big]^{\frac{\pi}{2}}_{\frac{\pi}{3}}$
$=-\Big[\cot\frac{\pi}{2}-\cot\frac{\pi}{2}\Big]$ $=-\Big[0-\frac{1}{\sqrt{3}}\Big]=+\frac{1}{\sqrt{3}}$

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 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

Suppose that $5 \%$ of men and $0.25 \%$ of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.

Solve the following equation for x:
$\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)+\cot^{-1}\Big(\frac{1-\text{x}^2}{2\text{x}}\Big)=\frac{2\pi}{3},\text{x}>0$

Let $f : R \rightarrow R$ be defined as $\text{f(x)}=\frac{2\text{x}-3}{4}.$ Write $fof^{-1}(1).$

A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15cm.

Write the coordinates of the point on the curve $\text{x}=\text{e}^\text{t}\cos\text{t},\text{y}=\text{e}^\text{t}$ where the tangent line makes an angle $\frac{\pi}{4}$ with x-axis.

Find the slope of the tangent to the curve $x = t^2 + 3t - 8, y = 2t^2 - 2t - 5$ at $t = 2.$

Find the mean and standard deviation of the following probability distributions:

$\text{x}_\text{i}$	$0$	$1$	$2$	$3$	$4$	$5$
$\text{p}_\text{i}$	$\frac{1}{6}$	$\frac{5}{18}$	$\frac{2}{9}$	$\frac{1}{6}$	$\frac{1}{9}$	$\frac{1}{18}$

Prove the following:
$\tan^{-1}\sqrt{x}=\frac{1}{2}\cos^{-1}\Bigg(\frac{1-x}{1+x}\Bigg),\text{x }\in(0, 1)$.

Evaluate the following definite integrals:
$\int_{1}^\limits{3}\frac{\log\text{x}}{(\text{x}+1)^2}\text{ dx}$

Write the multiplication table for the set of integers modulo $5.$