_ ^ e^ 2x 1 + 2x 1 + 2x ;dx = - Vidyadip

MCQ

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

A
${e^{2x}}\tan x + c$
B
${e^{2x}}\cot x + c$
✓
$\frac{{{e^{2x}}\tan x}}{2} + c$
D
$\frac{{{e^{2x}}\cot x}}{2} + c$

✓

Answer

Correct option: C.

$\frac{{{e^{2x}}\tan x}}{2} + c$

c
(c)$\int_{}^{} {{e^{2x}}\frac{{1 + \sin 2x}}{{1 + \cos 2x}}\,dx} = \int_{}^{} {{e^{2x}}\left[ {\frac{1}{{1 + \cos 2x}} + \frac{{\sin 2x}}{{1 + \cos 2x}}} \right]\,dx} $
$ = \int_{}^{} {{e^{2x}}\left[ {\frac{{{{\sec }^2}x}}{2} + \tan x} \right]} \,dx$
$ = \frac{1}{2}\int_{}^{} {{e^{2x}}{{\sec }^2}x\,dx} + \int_{}^{} {{e^{2x}}\tan x\,dx} $
$ = \frac{{{e^{2x}}\tan x}}{2} - \int_{}^{} {\frac{{{e^{2x}}{{\sec }^2}x}}{2}\,dx} + \int_{}^{} {\frac{{{e^{2x}}{{\sec }^2}x}}{2}\,dx} + c$
$ = \frac{{{e^{2x}}\tan x}}{2} + 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

More "M.C.Q (1 Marks)" from STD 12 - 7.1 inderfinite integral→STD 12 - 7.1 inderfinite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

The solution of differential equation $\frac{\text{dy}}{\text{dx}}-3\text{y}=\sin2\text{x}$ is:

The general solution of differention eqution of the type $\frac{\text{dx}}{\text{dy}}+\text{P}_{1}\text{x}=\text{Q}_{1}$ is:

If the point of minima of the function, $f(x) = 1 + a^2x -x^3 $ satisfy the inequality $\frac{{{x^2} + x + 2}}{{{x^2} + 5x + 6}} < 0$, then $'a'$ must lie in the interval

${d \over {dx}}({e^{{x^3}}})$ is equal to

Let $a = 2i + j - 2k$ and $b = i + j.$ If c is a vector such that $a\,.\,c = \,|c|,\,\,|c - a|\, = 2\sqrt 2 $and the angle between $(a \times b)$ and c is ${30^o}$, then $|\,(a \times b) \times c|\, = $

If $\text{A}=\begin{bmatrix} \text{a} & 0 & 0 \\ 0 & \text{a} & 0 \\ 0 & 0 &\text{a} \end{bmatrix},$ then the value of $|\text{adj A}|$ is:

If $\int\limits_{ - \infty }^\infty {f(x)dx = 1} $ then $\int\limits_{ - \infty }^\infty {f\left( {x - \frac{1}{x}} \right)dx} $ is equal to

If $\cos^{-1}\frac{\text{x}}{3}+\cos^{-1}\frac{\text{y}}{2}=\frac{\theta}{2},$ then, $4\text{x}^2-12\text{xy}\cos^2\frac{\theta}{2}+9\text{y}^2=$

Two ships $A$ and $B$ are sailing straight away from a fixed point $O$ along routes such that $\angle AOB$ is always $120^o$ . At a certain instance, $OA\, = 8\, km$, $OB\, = 6\, km$ and the ship $A$ is sailing at the rate of $20\, km/hr$ while the ship $B$ sailing at the rate of $30\, km/hr$. Then the distance between $A$ and $B$ is changing at the rate (in $km/hr$)

$\int_0^{\pi /4} {\frac{{4\sin 2\theta \,d\theta }}{{{{\sin }^4}\theta + {{\cos }^4}\theta }}} = $