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-x*exp(-x)

Integral of -x*exp(-x) dx

Limits of integration:

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The solution

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  1          
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 |      -x   
 |  -x*e   dx
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01xexdx\int\limits_{0}^{1} - x e^{- x}\, dx
Integral((-x)*exp(-x), (x, 0, 1))
Detail solution
  1. Use integration by parts:

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

    Let u(x)=xu{\left(x \right)} = - x and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{- x}.

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

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

    1. Let u=xu = - x.

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

      (eu)du\int \left(- e^{u}\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.

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

        So, the result is: eu- e^{u}

      Now substitute uu back in:

      ex- e^{- x}

    Now evaluate the sub-integral.

  2. Let u=xu = - x.

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

    (eu)du\int \left(- e^{u}\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.

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

      So, the result is: eu- e^{u}

    Now substitute uu back in:

    ex- e^{- x}

  3. Now simplify:

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

  4. 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             -x    -x
 | -x*e   dx = C + x*e   + e  
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xexdx=C+xex+ex\int - x e^{- x}\, dx = C + x e^{- x} + e^{- x}
The graph
0.001.000.100.200.300.400.500.600.700.800.902-1
The answer [src]
        -1
-1 + 2*e  
1+2e-1 + \frac{2}{e}
=
=
        -1
-1 + 2*e  
1+2e-1 + \frac{2}{e}
-1 + 2*exp(-1)
Numerical answer [src]
-0.264241117657115
-0.264241117657115
The graph
Integral of -x*exp(-x) dx

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