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

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

Limits of integration:

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The solution

You have entered [src] 
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∫−11xe−x2 dx\int\limits_{-1}^{1} x e^{- \frac{x}{2}}\, dx 
Integral(x*exp(-x/2), (x, -1, 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)=e−x2\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{2}}.

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

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

    1. Let u=−x2u = - \frac{x}{2}.

      Then let du=−dx2du = - \frac{dx}{2} and substitute −2du- 2 du:

      ∫(−2eu) du\int \left(- 2 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.

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

        So, the result is: −2eu- 2 e^{u}

      Now substitute uu back in:

      −2e−x2- 2 e^{- \frac{x}{2}}

    Now evaluate the sub-integral.

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

    ∫(−2e−x2) dx=−2∫e−x2 dx\int \left(- 2 e^{- \frac{x}{2}}\right)\, dx = - 2 \int e^{- \frac{x}{2}}\, dx

    1. Let u=−x2u = - \frac{x}{2}.

      Then let du=−dx2du = - \frac{dx}{2} and substitute −2du- 2 du:

      ∫(−2eu) du\int \left(- 2 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.

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

        So, the result is: −2eu- 2 e^{u}

      Now substitute uu back in:

      −2e−x2- 2 e^{- \frac{x}{2}}

    So, the result is: 4e−x24 e^{- \frac{x}{2}}

  3. Now simplify:

    −(2x+4)e−x2- \left(2 x + 4\right) e^{- \frac{x}{2}}

  4. Add the constant of integration:

    −(2x+4)e−x2+constant- \left(2 x + 4\right) e^{- \frac{x}{2}}+ \mathrm{constant}

The answer is:

−(2x+4)e−x2+constant- \left(2 x + 4\right) e^{- \frac{x}{2}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                 
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 |     2               2          2 
 | x*e    dx = C - 4*e    - 2*x*e   
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∫xe−x2 dx=C−2xe−x2−4e−x2\int x e^{- \frac{x}{2}}\, dx = C - 2 x e^{- \frac{x}{2}} - 4 e^{- \frac{x}{2}}
Simplify
The graph
-1.0-0.8-0.6-0.4-0.21.00.00.20.40.60.85-5
Plot the graph f(x)
The answer [src] 
     -1/2      1/2
- 6*e     + 2*e
−6e12+2e12- \frac{6}{e^{\frac{1}{2}}} + 2 e^{\frac{1}{2}} 
=
= 
     -1/2      1/2
- 6*e     + 2*e
−6e12+2e12- \frac{6}{e^{\frac{1}{2}}} + 2 e^{\frac{1}{2}} 
-6*exp(-1/2) + 2*exp(1/2)
Numerical answer [src] 
-0.341741416875544
-0.341741416875544

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

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