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x*e^(-x)dx

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

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1         
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 |  x*E   dx
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01exxdx\int\limits_{0}^{1} e^{- x} x\, dx
Integral(x*E^(-x), (x, 0, 1))
Detail solution
  1. Let u=xu = - x.

    Then let du=dxdu = - dx and substitute dudu:

    ueudu\int u e^{u}\, du

    1. Use integration by parts:

      udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

      Let u(u)=uu{\left(u \right)} = u and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

      Then du(u)=1\operatorname{du}{\left(u \right)} = 1.

      To find v(u)v{\left(u \right)}:

      1. The integral of the exponential function is itself.

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

      Now evaluate the sub-integral.

    2. The integral of the exponential function is itself.

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

    Now substitute uu back in:

    xexex- x e^{- x} - e^{- x}

  2. Now simplify:

    (x+1)ex- \left(x + 1\right) e^{- x}

  3. Add the constant of integration:

    (x+1)ex+constant- \left(x + 1\right) e^{- x}+ \mathrm{constant}


The answer is:

(x+1)ex+constant- \left(x + 1\right) e^{- x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                          
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 | x*E   dx = C - e   - x*e  
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exxdx=Cxexex\int e^{- x} x\, dx = C - x e^{- x} - e^{- x}
The graph
0.001.000.100.200.300.400.500.600.700.800.901-2
The answer [src]
       -1
1 - 2*e  
12e1 - \frac{2}{e}
=
=
       -1
1 - 2*e  
12e1 - \frac{2}{e}
1 - 2*exp(-1)
Numerical answer [src]
0.264241117657115
0.264241117657115
The graph
Integral of x*e^(-x)dx dx

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