Evaluate the following integrals: _ 0 ^ 2 1+ d - Vidyadip

Question

Evaluate the following integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$

✓

Answer

Let $\text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$
Let $\cos\theta=\text{t}$ Then, $-\sin\theta\text{ d}\theta=\text{dt}$
When $\theta=0,\text{t}=1$ and $\theta=\frac{\pi}{2},\text{t}=0$
$\therefore\ \text{I}=\int_{0}^\limits{\frac{\pi}{2}}\frac{\sin\theta}{\sqrt{1+\cos\theta}}\text{ d}\theta$
$=\int_{1}^\limits{0}\frac{-\text{dt}}{\sqrt{1+\text{t}}}$
$=\int_{0}^\limits{1}\frac{\text{dt}}{\sqrt{1+\text{t}}}$
$=2\big[\sqrt{1+\text{t}}\big]^1_0$
$=2\big(\sqrt{2}-1\big)$

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

Similar questions

Let A and B be two independent events such that $P(A) = p_1 $ and $P(B) = p_2.$ Describe in words the events whose probabilities are:
$1 - (1 - p_1)(1 - p_2).$

If $|\vec{\text{a}}|=\text{a}$ and $\big|\vec{\text{b}}\big|=\text{b},$ prove that $\Big(\frac{\vec{\text{a}}}{\text{a}^2}-\frac{\vec{\text{b}}}{\text{b}^2}\Big)^2=\Big(\frac{\vec{\text{a}}-\vec{\text{b}}}{\text{ab}}\Big)^2.$

If $\sin^{-1}\text{x}+\sin^{-1}\text{y}+\sin^{-1}\text{z}=\frac{3\pi}{2},$ then write the values of x + y + z.

Solve the following differential equation
$5\frac{\text{dy}}{\text{dx}}=\text{e}^\text{x}\text{y}^4$

A doctor claims that $60 \%$ of the patients he examines are allergic to some type of weed. What is the probability that exactly $3$ of his next $4$ patients are allergic to weeds?

Find the coordinates of the foot of the perpendicular drawn from the origin.
$x + y + z = 1$

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

If the sum of two unit vectors is a unit vector prove that the magnitude of their difference is $\sqrt{3.}$

Find mean $u,$variance $\sigma^{2}$for the following probability distribution:

X	0	1	2	3
P(X)	$\frac{1}{8}$	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{1}{8}$

Evaluate $\sin\Big(\tan^{-1}\frac{3}{4}\Big).$