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

Integral of (x^2)*exp(2x) dx

Limits of integration:

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The solution

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∫01x2e2x dx\int\limits_{0}^{1} x^{2} e^{2 x}\, dx 
Integral(x^2*exp(2*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)=x2u{\left(x \right)} = x^{2} and let dv⁡(x)=e2x\operatorname{dv}{\left(x \right)} = e^{2 x}.

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

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

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

      Now substitute uu back in:

      e2x2\frac{e^{2 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)=xu{\left(x \right)} = x and let dv⁡(x)=e2x\operatorname{dv}{\left(x \right)} = e^{2 x}.

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

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

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

      Now substitute uu back in:

      e2x2\frac{e^{2 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:

    ∫e2x2 dx=∫e2x dx2\int \frac{e^{2 x}}{2}\, dx = \frac{\int e^{2 x}\, dx}{2}

    1. Let u=2xu = 2 x.

      Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

      ∫eu2 du\int \frac{e^{u}}{2}\, 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: eu2\frac{e^{u}}{2}

      Now substitute uu back in:

      e2x2\frac{e^{2 x}}{2}

    So, the result is: e2x4\frac{e^{2 x}}{4}

  4. Now simplify:

    (2x2−2x+1)e2x4\frac{\left(2 x^{2} - 2 x + 1\right) e^{2 x}}{4}

  5. Add the constant of integration:

    (2x2−2x+1)e2x4+constant\frac{\left(2 x^{2} - 2 x + 1\right) e^{2 x}}{4}+ \mathrm{constant}

The answer is:

(2x2−2x+1)e2x4+constant\frac{\left(2 x^{2} - 2 x + 1\right) e^{2 x}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                        
 |                   2*x    2  2*x      2*x
 |  2  2*x          e      x *e      x*e   
 | x *e    dx = C + ---- + ------- - ------
 |                   4        2        2   
/
∫x2e2x dx=C+x2e2x2−xe2x2+e2x4\int x^{2} e^{2 x}\, dx = C + \frac{x^{2} e^{2 x}}{2} - \frac{x e^{2 x}}{2} + \frac{e^{2 x}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90010
Plot the graph f(x)
The answer [src] 
       2
  1   e 
- - + --
  4   4
−14+e24- \frac{1}{4} + \frac{e^{2}}{4} 
=
= 
       2
  1   e 
- - + --
  4   4
−14+e24- \frac{1}{4} + \frac{e^{2}}{4} 
-1/4 + exp(2)/4
Numerical answer [src] 
1.59726402473266
1.59726402473266

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

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