_0^ 2 e^ x/2 . ( x 2 + 4 ) ,dx = - Vidyadip

MCQ

$\int_0^{2\pi } {{e^{x/2}}.\sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right)\,dx = } $

A
$1$
B
$2\sqrt 2 $
✓
$0$
D
None of these

✓

Answer

Correct option: C.

$0$

c
(c) Let $I = \int_0^{2\pi } {{e^{x/2}}\sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right)\,dx} $

==> $I = 2\int_0^\pi {{e^t}\sin \left( {t + \frac{\pi }{4}} \right)dt} $

$= 2\left[ {\frac{{{e^t}}}{{\sqrt {1 + 1} }}\sin \left( {t + \frac{\pi }{4} - {{\tan }^{ - 1}}\frac{1}{1}} \right)} \right]_0^\pi $

$ = \frac{2}{{\sqrt 2 }}\left[ {{e^t}\sin t} \right]_0^\pi = \frac{2}{{\sqrt 2 }}[0] = 0$.

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 "M.C.Q (1 Marks)" from 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $f'(x) = {x^2} + 5$ and $f(0) = - 1$, then $f(x) = $

The domain of the function $f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$ )

Evaluate: $\int \frac{\cos x}{\left(\cos \frac{x}{2}+\sin \frac{x}{2}\right)^3} d x$

Let $\text{X}=\begin{bmatrix}\text{x}_1\\\text{x}_2\\\text{x}_3\end{bmatrix},\text{A}=\begin{bmatrix}1&-1&2\\2&0&1\\3&2&1\end{bmatrix}\text{and }\text{B}=\begin{bmatrix}3\\1\\4\end{bmatrix}$. If AX = B, then X is equal to:

$\begin{bmatrix}1\\2\\3\end{bmatrix}$
$\begin{bmatrix}-1\\-2\\-3\end{bmatrix}$
$\begin{bmatrix}-1\\2\\3\end{bmatrix}$
$\begin{bmatrix}0\\2\\1\end{bmatrix}$

If A is a square matrix such that A² = I, then A^-1 is equal to:

A + I
A
0
2A

The elimination of the arbitrary constants $A, B$ and $C$ from $y = A + Bx + C{e^{ - x}}$leads to the differential equation

If $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$, then $A$ is :

A five-digit number is written down at raddom. The probability that the number is divisible by 5, and no two consecutive digits are identical, is:

$\frac{1}{5}$
$\frac{1}{5}\big(\frac{9}{10}\big)^3$
$\big(\frac{3}{5}\big)^4$
$\text{None of these}$

$\int_{}^{} {\frac{{3{x^2}}}{{{x^6} + 1}}dx = } $

The difference of the order and the degree of the differential equation $\left(\frac{d^2 y}{d x^2}\right)^2+\left(\frac{d y}{d x}\right)^3+x^4=0$ is: