Solve: _ 0 ^ 2 1+ 2xdx = 1. 1 2 2. 1 3. 2 4. 3 2 - Vidyadip

Question

Solve: $\int\limits_{0}^{\frac{\pi}{2}}\sqrt{1+\sin2\text{x}\text{dx}}=$

$\frac{1}{2}$
$1$
$2$
$\frac{3}{2}$

✓

Answer

$2$

Solution:
$\int\limits_{0}^{\frac{\pi}{2}}\sqrt{1+\sin2\text{x}\text{dx}}=$
$\int\limits_{0}^{\frac{\pi}{2}}\sqrt{\sin^2\text{x}+\cos^2\text{x}+2\sin\text{x}\cos\text{xdx}}$
$\int\limits_{0}^{\frac{\pi}{2}}\sqrt{(\sin\text{x}+\cos\text{x})^2}\text{dx}$
$\int\limits_{0}^{\frac{\pi}{2}}(\sin\text{x}+\cos\text{x})\text{dx}$
$=[-\cos\text{x}+\sin\text{x}]\frac{\frac{\pi}{2}}{{0}}$
$=\Big[-\Big(\cos\frac{\pi}{2}-\cos0\Big)+\Big(\sin\frac{\pi}{2}-\sin0\Big)\Big]$
$=\big[-\big(0-1\big)+\big(1-0\big)\big]$
$=1+1=2$

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 "M.C.Q (1 Marks)" 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

What is the length of the longer diagonal of the parallelogram constructed on $5\vec{\text{a}}+2\vec{\text{b}}$ and $\vec{\text{a}}-3\vec{\text{b}}$ if it is given that $|\vec{\text{a}}|=2\sqrt{2},\big|\vec{\text{b}}\big|=3$ and the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ is $\frac{\pi}{4}$?

$15$
$\sqrt{113}$
$\sqrt{593}$
$\sqrt{369}$

Choose the correct answer from the given four options.
Let A and B be two events such that $\text{P}(\text{A})=\frac{3}{8},\text{P}({\text{B}})=\frac{5}{8}$ and $\text{P}(\text{A}\cup\text{B})=\frac{3}{4}.$Then $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)\cdot\text{P}\Big(\frac{\text{A'}}{\text{B}}\Big)$ is equal to:

Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when:

$\lambda>\frac{1}{2}$
$\lambda<\frac{1}{2}$
$\lambda<2$
$\lambda>2$

Let $\text{A}=\{\text{x}\in\text{R}:\text{x}\leq1\}$ and $f : A \rightarrow A$ be defined as $f(x) = x(2 - x).$ Then $f^{-1}(x)$ is$:$

If $\int\text{f(x)}\text{dx}=2\text{ f(x)}^3+\text{c}$ and $\text{f(x)}\neq0$ then f(x) is:

$\frac{\text{x}}{2}$
$\text{x}^3$
$\frac{1}{\sqrt{\text{x}}}$
$\sqrt{\frac{\text{x}}{3}}$

If $f: R \rightarrow R, f(x)=2 x+1$, then the value of $f^{-1}(5)$ is

The eqution of the plane contaning the two lines $\frac{\text{x}-1}{2}=\frac{\text{y}+1}{-1}=\frac{\text{z}-0}{3}$ and $\frac{\text{x}}{-2}=\frac{\text{y}-2}{-3}=\frac{\text{z}+1}{-1}$ is:

8x + y - 5z - 7 = 0
8x + y + 5z - 7 = 0
8x - y - 5z - 7 = 0
None of these

The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is:

If the function $f : R \rightarrow R$ be such that $f(x) = x - [x],$ where $[x]$ denotes the greatest integer less than or equal to $x,$ then $f^{-1}(x)$ is:

If $P(\operatorname{Not} A)=3 / 5$, then the value of $P(A)$ will be