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(x^3)exp^(x^2)

Solving integrals/ (x^3)exp^(-x^2)

Integral of (x^3)exp^(-x^2) dx

The solution

You have entered [src]

 oo           
  /           
 |            
 |        2   
 |   3  -x    
 |  x *E    dx
 |            
/             
0

$$\int\limits_{0}^{\infty} e^{- x^{2}} x^{3}\, dx$$

Integral(x^3*E^(-x^2), (x, 0, oo))

Detail solution

Let $u = - x^{2}$ .

Then let $du = - 2 x dx$ and substitute $\frac{du}{2}$ :

$\int \frac{u e^{u}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int u e^{u}\, du = \frac{\int u e^{u}\, du}{2}$
  1. Use integration by parts:
    
    $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
    
    Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$ .
    
    Then $\operatorname{du}{\left(u \right)} = 1$ .
    
    To find $v{\left(u \right)}$ :
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    Now evaluate the sub-integral.
  2. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  So, the result is: $\frac{u e^{u}}{2} - \frac{e^{u}}{2}$
Now substitute $u$ back in:

$- \frac{x^{2} e^{- x^{2}}}{2} - \frac{e^{- x^{2}}}{2}$
Now simplify:

$- \frac{\left(x^{2} + 1\right) e^{- x^{2}}}{2}$
Add the constant of integration:

$- \frac{\left(x^{2} + 1\right) e^{- x^{2}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{\left(x^{2} + 1\right) e^{- x^{2}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                     2         2
 |       2           -x     2  -x 
 |  3  -x           e      x *e   
 | x *E    dx = C - ---- - -------
 |                   2        2   
/

$$\int e^{- x^{2}} x^{3}\, dx = C - \frac{x^{2} e^{- x^{2}}}{2} - \frac{e^{- x^{2}}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

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

1/2

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

1/2

The graph

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

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Identical expressions

Similar expressions

Integral of (x^3)exp^(-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution