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Integral of d{x}:
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Integral of (x^5)e^(-x^2)
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Identical expressions
(x^ five)e^(-x^ two)
(x to the power of 5)e to the power of ( minus x squared )
(x to the power of five)e to the power of ( minus x to the power of two)
(x5)e(-x2)
x5e-x2
(x⁵)e^(-x²)
(x to the power of 5)e to the power of (-x to the power of 2)
x^5e^-x^2
(x^5)e^(-x^2)dx
Similar expressions
(x^5)e^(x^2)

Solving integrals/ (x^5)e^(-x^2)

Integral of (x^5)e^(-x^2) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |        2   
 |   5  -x    
 |  x *E    dx
 |            
/             
0

\int\limits_{0}^{1} e^{- x^{2}} x^{5}\, dx

Integral(x^5*E^(-x^2), (x, 0, 1))

Detail solution

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

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

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

$- \frac{x^{4} e^{- x^{2}}}{2} - x^{2} e^{- x^{2}} - e^{- x^{2}}$
Now simplify:

$- \left(\frac{x^{4}}{2} + x^{2} + 1\right) e^{- x^{2}}$
Add the constant of integration:

$- \left(\frac{x^{4}}{2} + x^{2} + 1\right) e^{- x^{2}}+ \mathrm{constant}$

The answer is:

- \left(\frac{x^{4}}{2} + x^{2} + 1\right) e^{- x^{2}}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                         
 |                                         2
 |       2             2         2    4  -x 
 |  5  -x            -x     2  -x    x *e   
 | x *E    dx = C - e    - x *e    - -------
 |                                      2   
/

\int e^{- x^{2}} x^{5}\, dx = C - \frac{x^{4} e^{- x^{2}}}{2} - x^{2} e^{- x^{2}} - e^{- x^{2}}

Simplify

The graph

Plot the graph f(x)

The answer [src]

       -1
    5*e  
1 - -----
      2

1 - \frac{5}{2 e}

       -1
    5*e  
1 - -----
      2

1 - \frac{5}{2 e}

1 - 5*exp(-1)/2

Numerical answer [src]

0.0803013970713942

0.0803013970713942

The graph

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

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Identical expressions

Similar expressions

Integral of (x^5)e^(-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution