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

Similar questions

Show the feasible region under the constraints :
$
\begin{aligned}
2 x+y & \leq 6 \\
x & \geq 0 \\
y & \geq 0
\end{aligned}
$

Find the principal values of the following:
$\text{cosec}^{-1}(-2)$

For what value of x, the following matrix is singular?
$\begin{vmatrix}5-\text{x}&\text{x}+1\\2&4\end{vmatrix}$

Find $\vec{\text{a}}.\vec{\text{b}}$ when
$\vec{\text{a}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=4\hat{\text{i}}-4\hat{\text{j}}+7\hat{\text{k}}$

Show that the function $f : R → {3} → R - {2}$ given by $\text{f(x)}=\frac{\text{x}-2}{\text{x}-3}$ is a bijection.

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 * be a binary operation on the set I of integers, defined by a * b = 2a + b - 3. Find the value of 3 * 4.

Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation give justification of this.
On $R,$ define by $a * b = ab^2.$
Here, $Z^+$ denotes the set of all non-negative integers.

For the matrix $\text{A}=\begin{bmatrix}1&5\\6&7\end{bmatrix}$, verify that

(A + A') is a symmetric matrix.
(A - A') is a skew symmentric matrix.

If $\begin{bmatrix}\text{x}+\text{y}\\\text{x}-\text{y} \end{bmatrix}=\begin{bmatrix}2&1\\4&3 \end{bmatrix}\begin{bmatrix}1\\-2\end{bmatrix},$ then write the value of (x, y).