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  • Identical expressions

  • exp(x^ two / two)*(-x)/ two
  • exponent of (x squared divide by 2) multiply by ( minus x) divide by 2
  • exponent of (x to the power of two divide by two) multiply by ( minus x) divide by two
  • exp(x2/2)*(-x)/2
  • expx2/2*-x/2
  • exp(x²/2)*(-x)/2
  • exp(x to the power of 2/2)*(-x)/2
  • exp(x^2/2)(-x)/2
  • exp(x2/2)(-x)/2
  • expx2/2-x/2
  • expx^2/2-x/2
  • exp(x^2 divide by 2)*(-x) divide by 2
  • exp(x^2/2)*(-x)/2dx

  • Similar expressions

  • exp(x^2/2)*(x)/2
Solving integrals/ exp(x^2/2)*(-x)/2

Integral of exp(x^2/2)*(-x)/2 dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |    2        
 |   x         
 |   --        
 |   2         
 |  e  *(-x)   
 |  -------- dx
 |     2       
 |             
/              
0
∫01−xex222 dx\int\limits_{0}^{1} \frac{- x e^{\frac{x^{2}}{2}}}{2}\, dx 
Integral((exp(x^2/2)*(-x))/2, (x, 0, 1))
Detail solution

  1. The integral of a constant times a function is the constant times the integral of the function:

    ∫−xex222 dx=∫(−xex22) dx2\int \frac{- x e^{\frac{x^{2}}{2}}}{2}\, dx = \frac{\int \left(- x e^{\frac{x^{2}}{2}}\right)\, dx}{2}

    1. The integral of a constant times a function is the constant times the integral of the function:

      ∫(−xex22) dx=−∫xex22 dx\int \left(- x e^{\frac{x^{2}}{2}}\right)\, dx = - \int x e^{\frac{x^{2}}{2}}\, dx

      1. There are multiple ways to do this integral.

        Method #1

        1. Let u=ex22u = e^{\frac{x^{2}}{2}}.

          Then let du=xex22dxdu = x e^{\frac{x^{2}}{2}} dx and substitute dudu:

          ∫1 du\int 1\, du

          1. The integral of a constant is the constant times the variable of integration:

            ∫1 du=u\int 1\, du = u

          Now substitute uu back in:

          ex22e^{\frac{x^{2}}{2}}

        Method #2

        1. Let u=x22u = \frac{x^{2}}{2}.

          Then let du=xdxdu = x dx and substitute dudu:

          ∫eu du\int e^{u}\, du

          1. The integral of the exponential function is itself.

            ∫eu du=eu\int e^{u}\, du = e^{u}

          Now substitute uu back in:

          ex22e^{\frac{x^{2}}{2}}

      So, the result is: −ex22- e^{\frac{x^{2}}{2}}

    So, the result is: −ex222- \frac{e^{\frac{x^{2}}{2}}}{2}

  2. Now simplify:

    −ex222- \frac{e^{\frac{x^{2}}{2}}}{2}

  3. Add the constant of integration:

    −ex222+constant- \frac{e^{\frac{x^{2}}{2}}}{2}+ \mathrm{constant}

The answer is:

−ex222+constant- \frac{e^{\frac{x^{2}}{2}}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                     
 |                      
 |   2                 2
 |  x                 x 
 |  --                --
 |  2                 2 
 | e  *(-x)          e  
 | -------- dx = C - ---
 |    2               2 
 |                      
/
∫−xex222 dx=C−ex222\int \frac{- x e^{\frac{x^{2}}{2}}}{2}\, dx = C - \frac{e^{\frac{x^{2}}{2}}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901.0-1.0
Plot the graph f(x)
The answer [src] 
     1/2
1   e   
- - ----
2    2
12−e122\frac{1}{2} - \frac{e^{\frac{1}{2}}}{2} 
=
= 
     1/2
1   e   
- - ----
2    2
12−e122\frac{1}{2} - \frac{e^{\frac{1}{2}}}{2} 
1/2 - exp(1/2)/2
Numerical answer [src] 
-0.324360635350064
-0.324360635350064

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

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