Evaluate the definite integral in Exercise: _0^ 4 2x dx - Vidyadip

Question

Evaluate the definite integral in Exercise:$\int\limits_0^{\frac{\pi}{4}}\sin2\text{x}\ \text{dx}$

✓

Answer

$\text{Let}\ \text{I}=\int\limits_{0}^{\frac{\text{x}}{4}}\sin2\text{x} \ \text{dx}$$\int\sin2\text{x}\ \text{dx}=\bigg(\frac{-\cos2\text{x}}{2}\bigg)=\text{F}\text{(x)}$
By second fundamental theorem of calculus, we obtain
$\text{I}=\text{F}\bigg(\frac{\pi}{4}\bigg)-\text{F}(0)$
$=-\frac{1}{2}\pi\bigg[\cos2\bigg(\frac{\pi}{4}\bigg)-\cos0\bigg] $
$=-\frac{1}{2}\bigg[\cos\bigg(\frac{\pi}{2}\bigg)-\cos0\bigg]$
$=-\frac{1}{2}[0-1]$
$=\frac{1}{2}$

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 Questions" from INTEGRALS→INTEGRALS — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following:$\big[2\hat{\text{i}}\hat{\text{j}}\hat{\text{k}}\big]+\big[\hat{\text{i}}\hat{\text{k}}\hat{\text{j}}\big]+\big[\hat{\text{k}}\hat{\text{j}}2\hat{\text{i}}\big]$

Find the area of the region bounded by the curve $y=\sin x$ and the $x$-axis where $0 \leq x \leq 2 \pi$.

Find the value of x, if:
$\begin{vmatrix}2&4\\5&1\end{vmatrix}=\begin{vmatrix}2\text{x}&4\\6&\text{x}\end{vmatrix}$

Find the vector equation of a plane which is at a distance of 3 units from the origin and has $\hat{\text{k}}$ as the unit vector normal to it.

Given two independent events A and B such that P(A) = 0.3 and P(B) = 0.6. Find
$\text{P}(\text{A}\cup\text{B})$

Define a differential equation.

Determine which of the following binary operations are associative and which are commutative:
'*' on N defined by a * b = 1 for all $\text{a, b}\in\text{N}.$

A discrete random variable $X$ has the probability distribution given below:

$X:$	$0.5$	$1$	$1.5$	$2$
$P(X):$	$k$	$k^2$	$2k^2$	$k$

Determine the mean of the distribution.

Find the area of the region bounded by $y=|x|$, $x =-3, x =1$ and the $x-$ axis.

Find the inverse of the matrix $\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}.$