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Integral of xexp(-x^2)
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Identical expressions
xexp(-x^ two)
x exponent of ( minus x squared )
x exponent of ( minus x to the power of two)
xexp(-x2)
xexp-x2
xexp(-x²)
xexp(-x to the power of 2)
xexp-x^2
xexp(-x^2)dx
Similar expressions
xexp(x^2)

Solving integrals/ xexp(-x^2)

Integral of xexp(-x^2) dx

The solution

You have entered [src]

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

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

Simplify

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = e^{- x^{2}}$ .
  
  Then let $du = - 2 x e^{- x^{2}} dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \frac{1}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $- \frac{u}{2}$
  Now substitute $u$ back in:
  
  $- \frac{e^{- x^{2}}}{2}$
Method #2
1. Let $u = - x^{2}$ .
  
  Then let $du = - 2 x dx$ and substitute $- \frac{du}{2}$ :
  
  $\int \frac{e^{u}}{4}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{e^{u}}{2}\right)\, du = - \frac{\int e^{u}\, du}{2}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $- \frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $- \frac{e^{- x^{2}}}{2}$
Add the constant of integration:

$- \frac{e^{- x^{2}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{e^{- x^{2}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                    
 |                    2
 |      2           -x 
 |    -x           e   
 | x*e    dx = C - ----
 |                  2  
/

$$-{{e^ {- x^2 }}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     -1
1   e  
- - ---
2    2

$${{1}\over{2}}-{{e^ {- 1 }}\over{2}}$$

=

     -1
1   e  
- - ---
2    2

$$- \frac{1}{2 e} + \frac{1}{2}$$

Simplify

Numerical answer [src]

0.316060279414279

0.316060279414279

The graph

Integral of xexp(-x^2) dx

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