_ ^ 1 + x 2 ;dx = - Vidyadip

MCQ

$\int_{}^{} {\sqrt {1 + \sin \frac{x}{2}} \;dx = } $

A
$\frac{1}{4}\left( {\cos \frac{x}{4} - \sin \frac{x}{4}} \right) + c$
B
$4\left( {\cos \frac{x}{4} - \sin \frac{x}{4}} \right) + c$
✓
$4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right) + c$
D
$4\left( {\sin \frac{x}{4} + \cos \frac{x}{4}} \right) + c$

✓

Answer

Correct option: C.

$4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right) + c$

c
(c)$\int_{}^{} {\sqrt {1 + \sin \frac{x}{2}\,} } dx = \int_{}^{} {\sqrt {\left( {{{\sin }^2}\frac{x}{4} + {{\cos }^2}\frac{x}{4} + 2\sin \frac{x}{4}\cos \frac{x}{4}} \right)} \,dx} $
$ = \int_{}^{} {\left( {\sin \frac{x}{4} + \cos \frac{x}{4}} \right)\,dx = 4\left( {\sin \frac{x}{4} - \cos \frac{x}{4}} \right)} + c$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If a committee of $3$ is to be chosen from a group of $38$ people of which you are a member. What is the probability that you will be on the committee

Let $A=\left(\begin{array}{cc}1 & 2 \\ -2 & -5\end{array}\right)$. Let $\alpha, \beta \in R$ be such that $\alpha A^{2}+\beta A=2 I$. Then $\alpha+\beta$ is equal to -

If the parabolas $y^2 = 4b\,(x -c)$ and $y^2 = 8ax$ have a common normal, then which one of the following is a valid choice for the ordered triad $(a, b, c)$ ?

The equation of the circle having as a diameter, the chord $x - y - 1 = 0$ of the circle $2{x^2} + 2{y^2} - 2x - 6y - 25 = 0$, is

Let $b$ be an non-zero real number. Suppose the quadratic equation $2 x^2+b x+\frac{1}{b}=0$ has two distinct real roots. Then

The $\operatorname{sum} \sum_{n=1}^{21} \frac{3}{(4 n-1)(4 n+3)}$ is equal to

If $a _{1}, a _{2}, a _{3} \ldots$ and $b _{1}, b _{2}, b _{3} \ldots$ are $A.P.$ and $a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}$ then $a_{4} b_{4}$ is equal to

Consider the system of equations $\mathrm{ax}+\mathrm{by}=0, \mathrm{cx}+\mathrm{dy}=0$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,1\}$.

$STATEMENT -1$ : The probability that the system of equations has a unique solution is $3 / 8$. and $STATEMENT - 2$: The probability that the system of equations has a solution is $1$ .

If the solution curve, of the differential equation $\frac{d y}{d x}=\frac{x+y-2}{x-y}$ passing through the point $(2,1)$ is $\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _e\left(\alpha+\left(\frac{y-1}{x-1}\right)^2\right)=\log _e|x-1|$, then $5 \beta+\alpha$ is equal to

From which of the following the distance of the point $(1,\,\,2,\,\,3)$ is $\sqrt {10} $