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xe^(x^ two - two)
xe to the power of (x squared minus 2)
xe to the power of (x to the power of two minus two)
xe(x2-2)
xex2-2
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xe^x^2-2
xe^(x^2-2)dx
Similar expressions
xe^(x^2+2)

Solving integrals/ xe^(x^2-2)

Integral of xe^(x^2-2) dx

The solution

You have entered [src]

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

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

Simplify

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x e^{x^{2} - 2} = \frac{x e^{x^{2}}}{e^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x e^{x^{2}}}{e^{2}}\, dx = \frac{\int x e^{x^{2}}\, dx}{e^{2}}$
  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:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      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:
    
    $\frac{e^{x^{2}}}{2}$
  So, the result is: $\frac{e^{x^{2}}}{2 e^{2}}$
Method #2
1. Rewrite the integrand:
  
  $x e^{x^{2} - 2} = \frac{x e^{x^{2}}}{e^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{x e^{x^{2}}}{e^{2}}\, dx = \frac{\int x e^{x^{2}}\, dx}{e^{2}}$
  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:
      
      $\int \frac{1}{2}\, du = \frac{\int 1\, du}{2}$
      
      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:
    
    $\frac{e^{x^{2}}}{2}$
  So, the result is: $\frac{e^{x^{2}}}{2 e^{2}}$
Now simplify:

$\frac{e^{x^{2} - 2}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

 -1    -2
e     e  
--- - ---
 2     2

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

 -1    -2
e     e  
--- - ---
 2     2

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

Simplify

Numerical answer [src]

0.116272078967415

0.116272078967415

The graph

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

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Identical expressions

Similar expressions

Integral of xe^(x^2-2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2