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

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

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∫01xxe(−1)x2 dx\int\limits_{0}^{1} x x e^{\frac{\left(-1\right) x}{2}}\, dx 
Integral((x*x)*exp((-x)/2), (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)=x2u{\left(x \right)} = x^{2} and let dv⁡(x)=e−x2\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{2}}.

    Then du⁡(x)=2x\operatorname{du}{\left(x \right)} = 2 x.

    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. 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)=−4xu{\left(x \right)} = - 4 x and let dv⁡(x)=e−x2\operatorname{dv}{\left(x \right)} = e^{- \frac{x}{2}}.

    Then du⁡(x)=−4\operatorname{du}{\left(x \right)} = -4.

    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.

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

    ∫8e−x2 dx=8∫e−x2 dx\int 8 e^{- \frac{x}{2}}\, dx = 8 \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: −16e−x2- 16 e^{- \frac{x}{2}}

  4. Now simplify:

    −(2x2+8x+16)e−x2- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}

  5. Add the constant of integration:

    −(2x2+8x+16)e−x2+constant- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}+ \mathrm{constant}

The answer is:

−(2x2+8x+16)e−x2+constant- \left(2 x^{2} + 8 x + 16\right) e^{- \frac{x}{2}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                
 |                                                 
 |      -x               -x         -x          -x 
 |      ---              ---        ---         ---
 |       2                2          2       2   2 
 | x*x*e    dx = C - 16*e    - 8*x*e    - 2*x *e   
 |                                                 
/
∫xxe(−1)x2 dx=C−2x2e−x2−8xe−x2−16e−x2\int x x e^{\frac{\left(-1\right) x}{2}}\, dx = C - 2 x^{2} e^{- \frac{x}{2}} - 8 x e^{- \frac{x}{2}} - 16 e^{- \frac{x}{2}}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-2020
Plot the graph f(x)
The answer [src] 
         -1/2
16 - 26*e
16−26e1216 - \frac{26}{e^{\frac{1}{2}}} 
=
= 
         -1/2
16 - 26*e
16−26e1216 - \frac{26}{e^{\frac{1}{2}}} 
16 - 26*exp(-1/2)
Numerical answer [src] 
0.230202847471531
0.230202847471531
The graph
Integral of (x*x)*exp(-x/2) dx

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

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