_0^ [2 e^ -x ] , dx where [x] denotes the greatest integer functi… - Vidyadip

MCQ

$\int\limits_0^\infty {} [2 e^{-x}]\, dx$ where $[x]$ denotes the greatest integer function is

A
$0$
✓
$ln\, 2$
C
$e^2$
D
$2/e$

✓

Answer

Correct option: B.

$ln\, 2$

b
for $0 < x < ln\, 2,\, [2e^{-x}] = 1,$ otherwise zero $ \Rightarrow I = \int\limits_0^{\ell n\,2} {} \,dx + \int\limits_{\ell n\,2}^\infty {} 0 \,dx = ln\, 2$

alternatively, put $e^{-x} = t ; - x = ln\, t ; dx = \frac{1}{t}\,dt$
$\int\limits_0^1 {[2t]\frac{1}{t}dt} $ $-\int\limits_0^1 {[2t]\frac{{dt}}{t}} $ ;

$\int\limits_0^{\frac{1}{2}} {0\,dt} $ $+\int\limits_{\frac{1}{2}}^{\frac{1}{2}} {\frac{{dt}}{t}} $

$=\left. {\ln \,t} \right]_{\frac{1}{2}}^{\,1}$

$= [1] - ln\, \frac{1}{2} = ln\, 2$

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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The differential equation for which ${\sin ^{ - 1}}x + {\sin ^{ - 1}}y = c$ is given by

If the system of equations $\mathrm{x}+4 \mathrm{y}-\mathrm{z}=\lambda$, $7 x+9 y+\mu z=-3,5 x+y+2 z=-1$ has infinitely many solutions, then $(2 \mu .+3 \lambda)$ is equal to :

The objective function of a linear programming problem is:

Choose the correct answer from the given four option.
Integrating factor of the differential equation $(1-\text{x}^2)\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{xy}=1$is:

$-\text{x}$
$\frac{\text{x}}{1+\text{x}^2}$
$\sqrt{1-\text{x}^2}$
$\frac{1}{2}\log(1-\text{x}^2)$

The element in the first row and third column of the inverse of the matrix $\left[ {\begin{array}{*{20}{c}}1&2&{ - 3}\\0&1&2\\0&0&1\end{array}} \right]$ is

Calculate: $\int(\text{x}^3-\frac{1}{\text{x}}+{3\text{x}})\text{dx:}$

$\frac{\text{x}^{4}}{3}-\log\text{x}+\frac{\text{5x}^{2}}{2}+\text{c}$
$\frac{\text{x}^{4}}{4}-\log\text{x}+\frac{\text{2x}^{2}}{3}+\text{c}$
$\frac{\text{x}^{4}}{4}-\log\text{x}+\frac{\text{3x}^{2}}{4}+\text{c}$
$\frac{\text{x}^{4}}{4}-\log\text{x}+\frac{\text{3x}^{2}}{2}+\text{c}$

If $S$ is the sum of the first $10$ terms of the series $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{7}\right)+\tan ^{-1}\left(\frac{1}{13}\right)+\tan ^{-1}\left(\frac{1}{21}\right)+\ldots$ then $\tan ( S )$ is equal to

The area bounded by curves $y = \cos x$ and $y = \sin x$ and ordinates $x = 0$ and $x = \frac{\pi }{4}$ is

If $f(x) = \cos x\cos 2x\cos 4x\cos 8x\cos 16x$, then $f'\left( {{\pi \over 4}} \right)$ is

The value of $ 'a' $ so that the volume of parallelopiped formed by $i + aj + k,j + a\,k$ and $a\,i + k$ becomes minimum is