Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^2
Integral of sin6xdx
Integral of xexp(x^2)
Integral of sqrt(5-x^2)
Identical expressions
e^(- zero .5x^ two)×x^ two
e to the power of ( minus 0.5x squared )×x squared
e to the power of ( minus zero .5x to the power of two)×x to the power of two
e(-0.5x2)×x2
e-0.5x2×x2
e^(-0.5x²)×x²
e to the power of (-0.5x to the power of 2)×x to the power of 2
e^-0.5x^2×x^2
e^(-0.5x^2)×x^2dx
Similar expressions
e^(0.5x^2)×x^2

Solving integrals/ e^(-0.5x^2)×x^2

Integral of e^(-0.5x^2)×x^2 dx

The solution

You have entered [src]

 oo            
  /            
 |             
 |     2       
 |   -x        
 |   ----      
 |    2    2   
 |  e    *x  dx
 |             
/              
0

$$\int\limits_{0}^{\infty} x^{2} e^{- \frac{x^{2}}{2}}\, dx$$

Simplify

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = x^{2}$ and let $\operatorname{dv}{\left(x \right)} = e^{- \frac{x^{2}}{2}}$ .

Then $\operatorname{du}{\left(x \right)} = 2 x$ .

To find $v{\left(x \right)}$ :
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \sqrt{2} \sqrt{\pi} x \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}\, dx = \sqrt{2} \sqrt{\pi} \int x \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x^{2} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\sqrt{2} x e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}} - \frac{\operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}$
So, the result is: $\sqrt{2} \sqrt{\pi} \left(\frac{x^{2} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2} + \frac{\sqrt{2} x e^{- \frac{x^{2}}{2}}}{2 \sqrt{\pi}} - \frac{\operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}\right)$
Now simplify:

$- x e^{- \frac{x^{2}}{2}} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}$
Add the constant of integration:

$- x e^{- \frac{x^{2}}{2}} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}+ \mathrm{constant}$

The answer is:

$- x e^{- \frac{x^{2}}{2}} + \frac{\sqrt{2} \sqrt{\pi} \operatorname{erf}{\left(\frac{\sqrt{2} x}{2} \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                
 |                                /                                              2 \                               
 |    2                           |     /    ___\         /    ___\            -x  |                      /    ___\
 |  -x                            |     |x*\/ 2 |    2    |x*\/ 2 |            ----|     ___   ____  2    |x*\/ 2 |
 |  ----                          |  erf|-------|   x *erf|-------|       ___   2  |   \/ 2 *\/ pi *x *erf|-------|
 |   2    2            ___   ____ |     \   2   /         \   2   /   x*\/ 2 *e    |                      \   2   /
 | e    *x  dx = C - \/ 2 *\/ pi *|- ------------ + --------------- + -------------| + ----------------------------
 |                                |       2                2                 ____  |                2              
/                                 \                                      2*\/ pi   /

$${{\sqrt{\pi}\,\mathrm{erf}\left({{x}\over{\sqrt{2}}}\right)}\over{ \sqrt{2}}}-x\,e^ {- {{x^2}\over{2}} }$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  ___   ____
\/ 2 *\/ pi 
------------
     2

$$\frac{\sqrt{2} \sqrt{\pi}}{2}$$

  ___   ____
\/ 2 *\/ pi 
------------
     2

$$\frac{\sqrt{2} \sqrt{\pi}}{2}$$

Simplify

The graph

Use the examples entering the upper and lower limits of integration.

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of e^(-0.5x^2)×x^2 dx

Limits of integration:

The graph:

Piecewise:

The solution