Download App

STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
$\int\limits_0^\infty  {} x^{2n + 1}·{e^{ - {x^2}}}\, dx$ is equal to $(n \in N).$
  • A
    $n !$
  • B
    $2 (n !)$
  • ✓
    $\frac{{n\,\,!}}{2}$
  • D
    $\frac{{(n + 1)!}}{2}\,$
✓

Answer

Correct option: C.
$\frac{{n\,\,!}}{2}$
c
$I = \int\limits_0^\infty  {{{({x^2})}^n}\,.\,x\,{e^{ - {x^2}}}\,\,dx\,} $

put $x^2 = t \Rightarrow x\, dx = - dt/2$
$=\frac{1}{2}\,\int\limits_0^\infty  {\,{t^n}\,\,{e^{ - t}}\,\,dt\,} $

$=\frac{1}{2}\,\left[ {\,\,{t^n}\,\left. {{e^{ - t}}} \right]_0^\infty \,\,\,\, + \,n\int\limits_0^\infty  {\,{t^{n - 1}}\,\,{e^{ - t}}\,\,dt\,} } \right]$ 

$= \frac{1}{2}\,\left[ {\,\,0\,\, + \,\,n\int\limits_0^\infty  {\,{t^{n - 1}}\,\,{e^{ - t}}\,\,dt\,} } \right]$
Hence $I = \frac{{n\,!\,}}{2}$

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 $\tan^{-1}(\cot\theta)=2\theta,$ then $\theta=$
→
If $\big|\vec{\text{a}}\times\vec{\text{b}}\big|=4,\big|\vec{\text{a}}.\vec{\text{b}}\big|=2,$ then $|\vec{\text{a}}|^2\big|\vec{\text{b}}\big|^2=$
→
If $a ^* b=a^2+b^2$, then the value of $(4 ^* 5) ^* 3$ is:
→
Given function $f(x) = \left( {{{{e^{2x}} - 1} \over {{e^{2x}} + 1}}} \right)$ is
→
Let the line of the shortest distance between the lines $L_1: \vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-\hat{j}+\hat{k})$ and  $L_2: \vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(\hat{i}+\hat{j}-\hat{k})$  intersect $\mathrm{L}_1$ and $\mathrm{L}_2$ at $\mathrm{P}$ and $\mathrm{Q}$  respectively. If $(\alpha, \beta, \gamma)$ is the midpoint  of the line segment $PQ$, then $2(\alpha+\beta+\gamma)$ is equal to____.
→
Let $\mathrm{x}^{\mathrm{k}}+\mathrm{y}^{\mathrm{k}}=\mathrm{a}^{\mathrm{k}},(\mathrm{a}, \mathrm{K}>0)$ and $\frac{\mathrm{dy}}{\mathrm{dx}}+\left(\frac{\mathrm{y}}{\mathrm{x}}\right)^{\frac{1}{3}}=0$ then $\mathrm{k}$ is
→
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors,then the greatest value of $\sqrt{3}\big|\vec{\text{a}}+\vec{\text{b}}\big|+\big|\vec{\text{a}}-\vec{\text{b}}\big|$ is:
→
The following system of equations $3x - 7y + 5z = 3; 3x + y + 5z = 7$ and $2x + 3y + 5z = 5$ are
→
The maximum value of $\sin x\,\,(1 + \cos x)$ will be at the
→
If $f : R \rightarrow R, g : R \rightarrow R$ and $h : R \rightarrow R$ are such that $f(x) = x^2$, $\text{g(x)}=\tan\text{x}$ and $\text{h(x)}=\log\text{x},$ then the value of $(go(foh)) (x),$ if $x = 1$ will be:
→