If f (x) = e^ -x + 2 e^ -2x + 3 e^ - 3x +...... + , then _ 2 ^ 3 … - Vidyadip

MCQ

If $f (x) = e^{-x} + 2 e^{-2x} + 3 e^{- 3x} +...... + \infty$ , then $\int\limits_{\ln 2}^{\ln 3} {f(x)\,dx} =$

A
$1$
✓
$\frac{1}{2}\,$
C
$\frac{1}{3}\,$
D
$ln\, 2$

✓

Answer

Correct option: B.

$\frac{1}{2}\,$

b
$\int\limits_{\ln 2}^{\ln 3} {} (e^{-x} + 2 e^{-2x} + 3 e^{- 3x} +...... + \infty ) dx$ $= - \left[ {{e^{ - x}}\,\, + \,\,{e^{ - 2x}}\,\, + \,{e^{ - 3x}}\, + {e^{ - 4x}}\, + \,......} \right]_{\ln 2}^{\ln 3}$
$= \left[ {\left\{ {\frac{1}{3}\, + \,\frac{1}{{{3^2}}}\,\, + \,\frac{1}{{{3^3}}}\,\, + .......} \right\}\,\,\, - \,\,\,\left\{ {\frac{1}{2}\, + \,\frac{1}{{{2^2}}}\,\, + \,\frac{1}{{{2^3}}}\,\, + .......} \right\}} \right]$
$= \frac{{1/2}}{{1 - 1/2}}\,\, = \,\,\frac{{1/3}}{{1 - 1/3}}\,\, = \,\,1 - \frac{1}{2}\,\, = \,\,\frac{1}{2}\,\,$

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 STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

If the position vectors of the points $A$ and $B$ are $i + 3j - k$ and $3i - j - 3k,$ then what will be the position vector of the mid point of $AB$

$\int\frac{1}{\cos\text{x}+\sqrt{3}\sin\text{x}}\text{ dx}$ is equal to:

If $f:R \to R$, is a continuous function such that $\left| {f\left( x \right) - f\left( y \right)} \right| \geqslant \left| {{e^x} - {e^y}} \right|\forall x,y \in R$ then $f(x)$ is

If $[x]$ is the greatest integer $\leq x$, then $\pi^{2} \int_{0}^{2}\left(\sin \frac{\pi \mathrm{x}}{2}\right)(\mathrm{x}-[\mathrm{x}])^{[\mathrm{x}]} \mathrm{d} \mathrm{x}$ is equal to :

The function $\text{f(x)}=|\cos\text{x}|$ is:

Let $P ( x )$ be a real polynomial of degree $3$ which vanishes at $x =-3 .$ Let $P ( x )$ have local minima at $x=1,$ local maxima at $x=-1$ and $\int_{-1}^{1} P ( x ) d x =18,$ then the sum of all the coefficients of the polynomial $P ( x )$ is equal to ....... .

Find the value of $ {\sin ^{ - 1}}\left( 1 \right):$

Given a system of inequatio$n:\ 2\text{y}-\text{x}\leq4$ $-2\text{x}+\text{y}\geq-4$.Find the value of $s,$ which is the greatest possible sum of the $x$ and $y\ co -$ ordinates of the point which satisfies the following inequalities as graphed in the $xy$ plane.

For which value of $x$, are the determinants $\left|\begin{array}{cc}2 x & -3 \\ 5 & x \end{array}\right|$ and $\left|\begin{array}{rr}10 & 1 \\ -3 & 2\end{array}\right|$ equal?

$\big(\vec{\text{a}}+2\vec{\text{b}}-\vec{\text{c}}\big).\big\{\big(\vec{\text{a}}-\vec{\text{b}}\big)\times\big(\vec{\text{a}}-\vec{\text{b}}-\vec{\text{c}}\big)\big\}$ is equal to: