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1/2xe^(-x/2)
Solving integrals/ 1/2xe^(-x/2)

Integral of 1/2xe^(-x/2) dx

Limits of integration:

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The solution

You have entered [src] 
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0
∫0∞xe(−1)x22 dx\int\limits_{0}^{\infty} \frac{x e^{\frac{\left(-1\right) x}{2}}}{2}\, dx
Simplify
Detail solution

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

    ∫xe(−1)x22 dx=∫xe(−1)x2 dx2\int \frac{x e^{\frac{\left(-1\right) x}{2}}}{2}\, dx = \frac{\int x e^{\frac{\left(-1\right) x}{2}}\, dx}{2}

    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. There are multiple ways to do this integral.

        Method #1

        1. Let u=−x2u = - \frac{x}{2}.

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

          ∫4eu du\int 4 e^{u}\, du

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

            ∫(−2eu) du=−2∫eu du\int \left(- 2 e^{u}\right)\, du = - 2 \int e^{u}\, du

            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}}

        Method #2

        1. Let u=e−x2u = e^{- \frac{x}{2}}.

          Then let du=−e−x2dx2du = - \frac{e^{- \frac{x}{2}} dx}{2} and substitute −2du- 2 du:

          ∫4 du\int 4\, du

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

            ∫(−2) du=−2∫1 du\int \left(-2\right)\, du = - 2 \int 1\, du

            1. The integral of a constant is the constant times the variable of integration:

              ∫1 du=u\int 1\, du = u

            So, the result is: −2u- 2 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:

        ∫4eu du\int 4 e^{u}\, du

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

          ∫(−2eu) du=−2∫eu du\int \left(- 2 e^{u}\right)\, du = - 2 \int e^{u}\, du

          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}}

    So, the result is: −xe−x2−2e−x2- x e^{- \frac{x}{2}} - 2 e^{- \frac{x}{2}}

  2. Now simplify:

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

  3. Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src] 
  /                               
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 |    -x                          
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 | x*e                 2        2 
 | ------ dx = C - 2*e    - x*e   
 |   2                            
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/
(−2 x−4) e−x22{{\left(-2\,x-4\right)\,e^ {- {{x}\over{2}} }}\over{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902-4
Plot the graph f(x)
The answer [src] 
2
22 
=
= 
2
22
Simplify
The graph
Integral of 1/2xe^(-x/2) dx

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

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