_ ^ dx e^ - 2x ( e^ 2x + 1) ^2 = - Vidyadip

MCQ

$\int_{}^{} {\frac{{dx}}{{{e^{ - 2x}}{{({e^{2x}} + 1)}^2}}} = } $

✓
$\frac{{ - 1}}{{2({e^{2x}} + 1)}} + c$
B
$\frac{1}{{2({e^{2x}} + 1)}} + c$
C
$\frac{1}{{{e^{2x}} + 1}} + c$
D
$\frac{{ - 1}}{{{e^{2x}} + 1}} + c$

✓

Answer

Correct option: A.

$\frac{{ - 1}}{{2({e^{2x}} + 1)}} + c$

a
(a)$\int_{}^{} {\frac{{dx}}{{{e^{ - 2x}}{{({e^{2x}} + 1)}^2}}}} = \int_{}^{} {\frac{{{e^{2x}}dx}}{{{{({e^{2x}} + 1)}^2}}}} $
Put $t = {e^{2x}} + 1 \Rightarrow \frac{{dt}}{2} = {e^{2x}}dx,$ then it reduces to
$\frac{1}{2}\int_{}^{} {\frac{1}{{{t^2}}}} \,dt = - \frac{1}{{2t}} + c = \frac{{ - 1}}{{2({e^{2x}} + 1)}} + c.$

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 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

Let $f, g: R \rightarrow R$ be two real valued functions defined as $f(x)=\left\{\begin{array}{cl}-|x+3| & , \quad x<0 \\ e^{x} & , \quad x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x & , \quad x<0 \\ 4 x+k_{2} & , \quad x \geq 0\end{array}\right.$, where $k_{1}$ and $k_{2}$ are real constants. If $(gof)$ is differentiable at $x=0$, then $(gof) (-4)+(gof)\, (4)$ is equal to

The line joining the points $6a - 4b + 4c,\, - 4c$ and the line joining the points $ - a - 2b - 3c,\,a + 2b - 5c$ intersect at

If two lines $L_1$ and $L_2$ in space, are defined by ${L_1} = \{ x = \sqrt \lambda y + \left( {\sqrt \lambda - 1} \right),z = \left( {\sqrt \lambda - 1} \right)y + \sqrt \lambda \} $ and ${L_2} = \{ x = \sqrt \mu y + \left( {1 - \sqrt \mu } \right),z = \left( {1 - \sqrt \mu } \right)y + \sqrt \mu \} $ then $L_1$ is perpendicular to $L_2$, for all non-negative reals $\lambda $ and $ \mu $, such that

If the position vectors of two point $ P $ and $Q $ are respectively $9i - j + 5k$ and $i + 3j + 5k$, and the line segment $PQ$ intersects the $ YOZ$ plane at a point $ R,$ the $PR : RQ$ is equal to

On the intervl $ (1,3)$, the function $f(x) = 3x + {2 \over x}$ is

Let $\overrightarrow{P R}=3 \hat{i}+\hat{j}-2 \hat{k}$ and $\overrightarrow{S Q}=\hat{i}-3 \hat{j}-4 \hat{k}$ determine diagonals of a parallelogram $P Q R S$ and $\overrightarrow{P T}=\hat{i}+2 \hat{j}+3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\overrightarrow{ PT }, \overrightarrow{ PQ }$ and $\overrightarrow{ PS }$ is

If the vectors represented by the sides $ AB $ and $BC$ of the regular hexagon $ABCDEF$ be $a$ and $ b$ , then the vector represented by $\overrightarrow {AE} $ will be

Let $R=\left(\begin{array}{lll}x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z\end{array}\right)$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta=y \sin \left(\theta+\frac{2 \pi}{3}\right)=z \sin \left(\theta+\frac{4 \pi}{3}\right)$ $\neq 0, \theta \in(0,2 \pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of M. Then, among the statements:

$(I)$ $Trace(\mathrm{R})=0$

$(II) $If $trace(\operatorname{adj}(\operatorname{adj}(\mathrm{R}))=0$, then $R$ has exactly one non-zero entry.

The vectors $a$ and $b$ are non-collinear. The value of $x$ for which the vectors $c = (x - 2)\,a + b$ and $d = (2x + 1)\,a - b$ are collinear, is

Let three vectors $\vec{a}, \overrightarrow{\mathrm{b}}$ and $\vec{c}$ be such that $\vec{a} \times \overrightarrow{\mathrm{b}}=\vec{c}, \overrightarrow{\mathrm{b}} \times \vec{c}=\vec{a}$ and $|\vec{a}|=2$

Then which one of the following is not true?