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Solving integrals/ x*e^(3-x^2)

Integral of x*e^(3-x^2) dx

The solution

You have entered [src]

  oo              
   /              
  |               
  |           2   
  |      3 - x    
  |   x*e       dx
  |               
 /                
  ___             
\/ 3

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $x e^{3 - x^{2}} = x e^{3} e^{- x^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int x e^{3} e^{- x^{2}}\, dx = e^{3} \int x e^{- x^{2}}\, dx$
  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:
      
      $\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$
      
      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: $- \frac{e^{3} e^{- x^{2}}}{2}$
Method #2
1. Rewrite the integrand:
  
  $x e^{3 - x^{2}} = x e^{3} e^{- x^{2}}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int x e^{3} e^{- x^{2}}\, dx = e^{3} \int x e^{- x^{2}}\, dx$
  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:
      
      $\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$
      
      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: $- \frac{e^{3} e^{- x^{2}}}{2}$
Now simplify:

$- \frac{e^{3 - x^{2}}}{2}$
Add the constant of integration:

$- \frac{e^{3 - x^{2}}}{2}+ \mathrm{constant}$

The answer is:

$- \frac{e^{3 - x^{2}}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$$-{{e^{3-x^2}}\over{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

$${{1}\over{2}}-{{e^{3-{\it oo}^2}}\over{2}}$$

1/2

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

Упростить

The graph

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

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Identical expressions

Similar expressions

Integral of x*e^(3-x^2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2