_ ^ x + x 1 + x ;dx is equal to

MCQ

$\int_{}^{} {\frac{{x + \sin x}}{{1 + \cos x}}\;dx} $ is equal to

A
$\frac{1}{2}x\tan \frac{x}{2} + c$
✓
$x\tan \;\frac{x}{2} + c$
C
$x\tan x + c$
D
$\frac{1}{2}x\tan x + c$

✓

Answer

Correct option: B.

$x\tan \;\frac{x}{2} + c$

b
(b)$\int_{}^{} {\frac{{x + \sin x}}{{1 + \cos x}}\,dx = \frac{1}{2}\int_{}^{} {x{{\sec }^2}\frac{x}{2}\,dx + \int_{}^{} {\tan \frac{x}{2}\,dx} } } $
$ = \frac{1}{2}\frac{{x\tan \frac{x}{2}}}{{\frac{1}{2}}} - \int_{}^{} {\tan \frac{x}{2}\,dx} + \int_{}^{} {\tan \frac{x}{2}\,dx} $$ = x\tan \frac{x}{2} + c$.
Trick : By inspection, $\frac{d}{{dx}}\left\{ {x\tan \frac{x}{2} + c} \right\}$
$ = \frac{x}{2}{\sec ^2}\frac{x}{2} + \tan \frac{x}{2} = \frac{1}{2}\left[ {\frac{x}{{{{\cos }^2}\frac{x}{2}}} + \frac{{2\sin \frac{x}{2}}}{{\cos \frac{x}{2}}}} \right] = \frac{{x + \sin x}}{{1 + \cos x}}$.

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 7.1 inderfinite integral→7.1 inderfinite 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 $\tan^{-1}(\cot\theta)=2\theta,$ then $\theta=$

$\pm\frac{\pi}{3}$
$\pm\frac{\pi}{4}$
$\pm\frac{\pi}{6}$
$\text{none of these}$

→

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb{R}$. Then

($A$) $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$

($B$) $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$

($C$) $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$

($D$) $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$

→

The production of item $A$ is $x$ and the production of item $B$ is $y .$ If the corner points of the bounded feasible region are $(1,0),(2,0),(0,2)$ and $(0,1)$ then the maximum profit $z=2000 x+5000 y$ is $\ldots \ldots$

→

Which of the following triplets give the direction cosines of a line:

1, 1, 1
1, -1, 1
1, 1, -1
$\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

→

Let $y=y(x)$ be the solution of the differential equation

$\frac{d y}{d x}=1+x e^{y-x},-\sqrt{2}\,<\,x\,<\,\sqrt{2}, y (0)=0$ then, the minimum value of $y(x)$ , $\mathrm{x} \in(-\sqrt{2}, \sqrt{2})$ is equal to:

→

The value of $\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} } \right)$ is

→

If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is: