_ - 1 ^0 4 x^2 + 4x + 3 1 + e^ 2x + 1 dx , = - Vidyadip

MCQ

$\int\limits_{ - 1}^0 {\frac{{4{x^2} + 4x + 3}}{{1 + {e^{2x + 1}}}}} dx\, = $

A
$\frac{7}{3}$
B
$0$
✓
$\frac{7}{6}$
D
$\frac{7}{12}$

✓

Answer

Correct option: C.

$\frac{7}{6}$

c
$\int_{-1}^{0} \frac{(2 x+1)^{2}+2}{1+e^{2 x+1}}$

$\text { Put } 2 x+1=t, d x=\frac{d t}{2}$

$I=\frac{1}{2} \int_{-1}^{1} \frac{t^{2}+2}{1+e^{t}}$

$2 \mathrm{I}=\frac{1}{2} \int_{-1}^{1}\left(\mathrm{t}^{2}+2\right) \mathrm{dt}$

$2 \mathrm{I}=\frac{1}{2} \times 2 \int_{0}^{1}\left(\mathrm{t}^{2}+2\right) \mathrm{dt}$

$2 \mathrm{I}=\left(\frac{t^{3}}{3}+2 t\right)_{0}^{1}$

$I=\frac{7}{6}$

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\lim _{x \rightarrow 0} \frac{ ae ^{x}- b \cos x + ce ^{- x }}{ x \sin x }=2,$ then $a + b + c$ is equal to ...........

Consider the function $f (x) =$$\left[ \begin{gathered} \hfill \\ \hfill \\ \hfill \\ \hfill \\ \hfill \\ \end{gathered} \right.$$\begin{gathered} \frac{x}{{[x]}}\;\;\;\;\;\;\;\;\;\;\;\;\;if\;\;1 \leqslant \;x < 2 \hfill \\ \hfill \\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;if\;\;x = 2 \hfill \\ \hfill \\ \sqrt {6 - x} \;\;\;\;\;\;\;if\;\;2 < x \leqslant 3 \hfill \\ \end{gathered} $

where $[x]$ denotes step up function then at $x = 2$ function

An experiment has $10$ equally likely outcomes. Let $\mathrm{A}$ and $\mathrm{B}$ be two non-empty events of the experiment. If $\mathrm{A}$ consists of $4$ outcomes, the number of outcomes that $B$ must have so that $A$ and $B$ are independent, is

In a single throw of two dice the probability of obtaining an odd number is

The number of real solutions of the equation

$\sin ^{-1}\left(\sum_{i=1}^{\infty} x^{i+1}-x \sum_{i=1}^{\infty}\left(\frac{x}{2}\right)^i\right)=\frac{\pi}{2}-\cos ^{-1}\left(\sum_{i=1}^{\infty}\left(-\frac{x}{2}\right)^i-\sum_{i=1}^{\infty}(-x)^i\right)$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is. . . . .

(Here, the inverse trigonometric functions $\sin ^{-1} x$ and $\cos ^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

If $f(x)=\left|\begin{array}{ccc}x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2\end{array}\right|$ for all $x \in \mathbb{R}$, then $2 f(0)+f^{\prime}(0)$ is equal to

Coasider a cuboid of sides $2 x , 4 x$ and $5 x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is a constant $k$, then the ratio $x: r$, for which the sum of their volumes is maximum, is

If the circles $x^{2}+y^{2}+6 x+8 y+16=0$ and $x^{2}+y^{2}+2(3-\sqrt{3}) x+x+2(4-\sqrt{6}) y$ $= k +6 \sqrt{3}+8 \sqrt{6}, k >0$, touch internally at the point $P(\alpha, \beta)$, then $(\alpha+\sqrt{3})^{2}+(\beta+\sqrt{6})^{2}$ is equal to $\dots\dots$

Let the function $\mathrm{g}:(-\infty, \infty) \rightarrow\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ be given by $g(\mathrm{u})=2 \tan ^{-1}\left(e^{\mathrm{u}}\right)-\frac{\pi}{2}$. Then, $\mathrm{g}$ is

The centre of the circle $x = 2 + 3\cos \theta $, $y = 3\sin \theta - 1$ is