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

Integral of -x*exp(x) dx

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The solution

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∫01−xex dx\int\limits_{0}^{1} - x e^{x}\, dx 
Integral((-x)*exp(x), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\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. The integral of the exponential function is itself.

      ∫ex dx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

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

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

    1. The integral of the exponential function is itself.

      ∫ex dx=ex\int e^{x}\, dx = e^{x}

    So, the result is: −ex- e^{x}

  3. Now simplify:

    (1−x)ex\left(1 - x\right) e^{x}

  4. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
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∫−xex dx=C−xex+ex\int - x e^{x}\, dx = C - x e^{x} + e^{x}
Simplify
The graph
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The answer [src] 
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Numerical answer [src] 
-1.0
-1.0
The graph
Integral of -x*exp(x) dx

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

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