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x^2*exp(-x^2)
Identical expressions
x^ two *exp(-x^ two)
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x to the power of two multiply by exponent of ( minus x to the power of two)
x2*exp(-x2)
x2*exp-x2
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x^2exp(-x^2)
x2exp(-x2)
x2exp-x2
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x^2*exp(-x^2)dx
Similar expressions
x^2*exp(-x^2/2)
x^2*exp(x^2)
(x^2)*exp(-x^2)

Solving integrals/ x^2*exp(-x^2)

Integral of x^2*exp(-x^2) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |        2   
 |   2  -x    
 |  x *e    dx
 |            
/             
0

$$\int\limits_{0}^{1} x^{2} e^{- x^{2}}\, dx$$

Integral(x^2*exp(-x^2), (x, 0, 1))

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 $\operatorname{dv}{\left(x \right)} = e^{- x^{2}}$ .

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:

$\int \sqrt{\pi} x \operatorname{erf}{\left(x \right)}\, dx = \sqrt{\pi} \int x \operatorname{erf}{\left(x \right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x^{2} \operatorname{erf}{\left(x \right)}}{2} + \frac{x e^{- x^{2}}}{2 \sqrt{\pi}} - \frac{\operatorname{erf}{\left(x \right)}}{4}$
So, the result is: $\sqrt{\pi} \left(\frac{x^{2} \operatorname{erf}{\left(x \right)}}{2} + \frac{x e^{- x^{2}}}{2 \sqrt{\pi}} - \frac{\operatorname{erf}{\left(x \right)}}{4}\right)$
Now simplify:

$- \frac{x e^{- x^{2}}}{2} + \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{4}$
Add the constant of integration:

$- \frac{x e^{- x^{2}}}{2} + \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{x e^{- x^{2}}}{2} + \frac{\sqrt{\pi} \operatorname{erf}{\left(x \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                            
 |                         /                             2 \                   
 |       2                 |            2              -x  |     ____  2       
 |  2  -x             ____ |  erf(x)   x *erf(x)    x*e    |   \/ pi *x *erf(x)
 | x *e    dx = C - \/ pi *|- ------ + --------- + --------| + ----------------
 |                         |    4          2           ____|          2        
/                          \                       2*\/ pi /

$$\int x^{2} e^{- x^{2}}\, dx = C + \frac{\sqrt{\pi} x^{2} \operatorname{erf}{\left(x \right)}}{2} - \sqrt{\pi} \left(\frac{x^{2} \operatorname{erf}{\left(x \right)}}{2} + \frac{x e^{- x^{2}}}{2 \sqrt{\pi}} - \frac{\operatorname{erf}{\left(x \right)}}{4}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   -1     ____       
  e     \/ pi *erf(1)
- --- + -------------
   2          4

$$- \frac{1}{2 e} + \frac{\sqrt{\pi} \operatorname{erf}{\left(1 \right)}}{4}$$

   -1     ____       
  e     \/ pi *erf(1)
- --- + -------------
   2          4

$$- \frac{1}{2 e} + \frac{\sqrt{\pi} \operatorname{erf}{\left(1 \right)}}{4}$$

Упростить

Numerical answer [src]

0.189472345820492

0.189472345820492

The graph

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

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Identical expressions

Similar expressions

Integral of x^2*exp(-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution