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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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∫01xex 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 the exponential function is itself.

    ∫ex dx=ex\int e^{x}\, dx = 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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∫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] 
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Integral of x*exp(x) dx

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

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