^ _01 1+e^x dx equals: 1. 2-1 2. 2 3. 4-1 4. - 2 - Vidyadip

Question

$\int\limits^\infty_0\frac{1}{1+\text{e}^\text{x}}\text{dx}$ equals:

$\log2-1$
$\log2$
$\log4-1$
$-\log2$

✓

Answer

$\log\Big(\frac{4}{3}\Big)$

Solution:

Let, $\text{I}=\int\limits^\frac{\pi}{2}_0\frac{\cos\text{x}}{(2+\sin\text{x})(1+\sin\text{x})}\text{dx}$

Let $\sin\text{x}$ then $\cos\text{x}\text{ dx}=\text{dt}$

When $\text{x}=0,\text{t}=0,\text{x}=\frac{\pi}{2},\text{t}=1$

Therefore the integral becomes

$\text{I}=\int\limits^1_0\frac{\text{dt}}{(2+\text{t})(1+\text{t})}$

$=\int\limits^1_0\Big[\frac{-1}{2+\text{t}}+\frac{1}{1+\text{t}}\Big]\text{dt}$

$=\big[-\log(2+\text{t})+\log(1+\text{t})\big]^1_0$

$= \big[\log(1+\text{t})-\log(2+\text{t})\big]^1_0$

$=\log2-\log3-\log1+\log2$

$=\log\frac{4}{3}$

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 Definite Integrals→Definite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If A is an invertible matrix, then which of the following is not true:

$(\text{A}^2)^\text{-1}=(\text{A}^{-1})^2$
$|\text{A}^{-1}|=|\text{A}|^{-1}$
$(\text{A}^\text{T})^\text{-1}=(\text{A}^{-1})^\text{T}$
$|\text{A}|\neq0$

If $f(x) = \left\{ {\begin{array}{*{20}{c}}{\frac{{{x^2} - 9}}{{x - 3}}\,,}&{{\rm{if \,\,}}x \ne 3}\\{2x + k\,,}&{{\rm{otherwise}}}\end{array}} \right.$, is continuous at $x = 3,$ then $k = $

If $\text{A}=\begin{bmatrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{bmatrix},$ then det (adj (adj A)) is:

14⁴
14³
14²
14

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

Choose the correct answer from the given four options.
Let f : R → R be defined by $\text{f}(\text{x})=\frac{1}{\text{x}}\ \forall\ \text{x}\in\text{R}.$ Then f is:

one-one.
onto.
bijective.
f is not defined.

If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\sqrt {1 - x} \,\,\,\,\,\,\,\,\,\,\,\,\,0 \le x \le 1}\\
{{{(7x - 6)}^{\frac{{ - 1}}{3}}}\,\,\,\,\,1 < x \le 2}
\end{array}} \right.$ , then $\int\limits_0^2 {{{f}}(x)\,dx} $ is equal to

If $a = i - 2j + 3k$ and $b = 3i + j + 2k,$ then the unit vector perpendicular to $a$ and $b$ is

A tetrahedron has vertices at $O(0,\,0,\,0)$, $A(1,\,2,\,1),B(2,\,1,\,3)$ and $C( - 1,\,1,\,2)$. Then the angle between the faces $OAB$ and $ABC$will be

Let the functions $:(-1,1) \rightarrow R$ and $g:(-1,1) \rightarrow(-1,1)$ be defined by $f(x)=|2 x-1|+|2 x+1|$ and $g(x)=x-[x]$,

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ:(-1,1) \rightarrow R$ be the composite function defined by $(f \circ g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is $NOT$ differentiable. Then the value of $c+d$ is. . . . .

Suppose that $f$ is differentiable for all $x$ and that $f '(x) \le 2$ for all x. If $f (1) = 2$ and $f (4) = 8$ then $f (2)$ has the value equal to