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

Question

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\sqrt{1+\cos\text{x}}\text{ dx}$

✓

Answer

We use $1+\cos\text{x}=2\cos^2\frac{\text{x}}{2}$
$=\int_{0}^\limits{\frac{\pi}{2}}\sqrt{2\cos^2\frac{\text{x}}{2}}\text{ dx}$
$=\int_{0}^\limits{\frac{\pi}{2}}\sqrt{2}\cos\frac{\text{x}}{2}\text{ dx}$
$=\sqrt{2}\Big[2\sin\frac{\text{x}}{2}\Big]^{\frac{\pi}{2}}_0$
$=2\sqrt{2}\Big[\frac{1}{\sqrt{2}}\Big]$
$=2$

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 "2 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 value of $\sec ^{-1}(-2)-\sin ^{-1}\left(\frac{1}{2}\right)$.

Evaluate the following determinant:
$\begin{vmatrix}1&-3&2\\4&-1&2\\3&5&2\end{vmatrix}$

Find the adjoint of the following matrices:

$\text{C}=\begin{bmatrix} \cos\alpha & \sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$

Verify that (adjoint A) A = |A|I = A (adjoint A) for the above matrices.

If in a binomial distribution mean is 5 and variance is 4, write the number of trials.

If A is a non-singular square matrix such that |A| = 10, find |A^-1|.

$\text{If}\ \vec{a}\cdot\vec{a}=0\ \text{and}\ \vec{a}\cdot\vec{b}=0,$ then what can be concluded about the vector $\vec{b}?$

If the position vector of a point (-4, -3) be $\vec{\text{a}}$, find $\big|\vec{\text{a}}\big|$.

Write the value of $\big(\vec{\text{a}}.\hat{\text{i}}\big)\hat{\text{i}}+\big(\vec{\text{a}}.\hat{\text{j}}\big)\hat{\text{j}}+\big(\vec{\text{a}}.\hat{\text{k}}\big)\hat{\text{k}},$ where $\vec{\text{a}}$ is any vector.

Let A be a square matrix such that A² - A + I = 0, then write A^-1 interms of A.

Construct a 2 × 3 matrix A = [a_ij] whose elements a_ij are give by:
a_ij = i.j