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

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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
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∫01xe−x2 dx\int\limits_{0}^{1} x e^{- x^{2}}\, dx 
Integral(x*exp(-x^2), (x, 0, 1))
Detail solution

  1. Let u=−x2u = - x^{2}.

    Then let du=−2xdxdu = - 2 x dx and substitute −du2- \frac{du}{2}:

    ∫(−eu2) du\int \left(- \frac{e^{u}}{2}\right)\, du

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

      False\text{False}

      1. The integral of the exponential function is itself.

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

      So, the result is: −eu2- \frac{e^{u}}{2}

    Now substitute uu back in:

    −e−x22- \frac{e^{- x^{2}}}{2}

  2. Add the constant of integration:

    −e−x22+constant- \frac{e^{- x^{2}}}{2}+ \mathrm{constant}

The answer is:

−e−x22+constant- \frac{e^{- x^{2}}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                    
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∫xe−x2 dx=C−e−x22\int x e^{- x^{2}}\, dx = C - \frac{e^{- x^{2}}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.901-1
Plot the graph f(x)
The answer [src] 
     -1
1   e  
- - ---
2    2
12−12e\frac{1}{2} - \frac{1}{2 e} 
=
= 
     -1
1   e  
- - ---
2    2
12−12e\frac{1}{2} - \frac{1}{2 e} 
1/2 - exp(-1)/2
Numerical answer [src] 
0.316060279414279
0.316060279414279
The graph
Integral of x*exp(-x^2) dx

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

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