_0^ /2 1 + 2 x (2 + x) ^2 = - Vidyadip

MCQ

$\int_0^{\pi /2} {\frac{{1 + 2\cos x}}{{{{(2 + \cos x)}^2}}} = } $

A
$\frac{\pi }{2}$
B
$\pi $
✓
$\frac{1}{2}$
D
None of these

✓

Answer

Correct option: C.

$\frac{1}{2}$

c
(c) $\int_0^{\pi /2} {\frac{{(1 + 2\cos x)}}{{{{(2 + \cos x)}^2}}}dx = \int_0^{\pi /2} {\frac{{2(\cos x + 2) - 3}}{{{{(2 + \cos x)}^2}}}dx} } $

$ = 2\int_0^{\pi /2} {\frac{{dx}}{{2 + \cos x}} - 3\int_0^{\pi /2} {\frac{{dx}}{{{{(2 + \cos x)}^2}}}} } $

$ = 4\int_0^1 {\frac{{dt}}{{3 + {t^2}}} - 6\int_0^1 {\frac{{1 + {t^2}}}{{{{(3 + {t^2})}^2}}}dt} } $, $\left[ {{\rm{Put}}\,\,\tan \frac{x}{2} = t} \right]$

$ = - 2\int_0^1 {\frac{{dt}}{{3 + {t^2}}} + 12\int_0^1 {\frac{{dt}}{{{{(3 + {t^2})}^2}}}} } $

$ = - 2\int_0^1 {\frac{{dt}}{{3 + {t^2}}} + 12} \left[ {\frac{1}{6}.\frac{t}{{{t^2} + 3}}} \right]_0^1 + \frac{1}{6}\int_0^1 {\frac{{dt}}{{3 + {t^2}}}} $

$ = 2\left[ {\frac{t}{{{t^2} + 3}}} \right]_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 "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

The value of $\big[\vec{\text{a}}-\vec{\text{b}},\vec{\text{b}}-\vec{\text{c}},\vec{\text{c}}-\vec{\text{a}}\big],$ where $\big|\vec{\text{a}}\big|=1,\big|\vec{\text{b}}\big|=5,\big|\vec{\text{c}}\big|=3,$ is:

0
1
6
None of these.

The value of the integral $\int \limits_{-\pi / 2}^{\pi / 2} \frac{d x}{\left(1+e^{x}\right)\left(\sin ^{6} x+\cos ^{6} x\right)}$ is equal to

Let $D_1 =$ $\left| {\,\begin{array}{*{20}{c}}a&b&{a + b}\\c&d&{c + d}\\a&b&{a - b}\end{array}\,} \right|$ and $D_2 =$ $\left| {\,\begin{array}{*{20}{c}}a&c&{a + c}\\b&d&{b + d}\\a&c&{a + b + c}\end{array}\,} \right|$ then the value of $\frac{{{D_1}}}{{{D_2}}}$ where $b \ne 0$ and $ad \ne bc$, is

Which of the following is an essential condition in a situation for linear programming to be useful?

Choose the correct answer from the given four options.
The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is:

$-\hat{\text{i}}+12\hat{\text{j}}+4\hat{\text{k}}$
$-5\hat{\text{i}}+2\hat{\text{j}}-4\hat{\text{k}}$
$-5\hat{\text{i}}+12\hat{\text{j}}+4\hat{\text{k}}$
$\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$

If $\text{A}=\begin{bmatrix}\alpha&\beta\\\gamma&-\alpha\end{bmatrix}$ is such that A² = I, then:

$1+\alpha^2+\beta\gamma=0$
$1-\alpha^2+\beta\gamma=0$
$1-\alpha^2-\beta\gamma=0$
$1+\alpha^2-\beta\gamma=0$

Let $y=y(x)$ be the solution of the differential equation $\frac{d y}{d x}=2 x(x+y)^3-x(x+y)-1, y(0)=1$. Then, $\left(\frac{1}{\sqrt{2}}+\mathrm{y}\left(\frac{1}{\sqrt{2}}\right)\right)^2$ equals : . . . . . . .

$\int\text{x}^{\sin\text{x}}\Big(\frac{\sin\text{x}}{\text{x}}+\cos\text{x}\cdot\log\text{x}\Big)\text{dx}$ is equal to:

$\text{x}^{\sin\text{x}}+\text{C}$
$\text{x}^{\sin\text{x}}\cos\text{x}+\text{C}$
$\frac{(\text{x}^{\sin\text{x}})^2}{2}+\text{C}$
None of these.

Let $f(x) = sin\ x, g(x) = cos\ x$ , then which of the following statement is false

The order of any matrix is 3 × 2 then no.of element in the matrix:

3
2
5
6