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x^2*exp(3*x)
Solving integrals/ x^2*exp(3*x)

Integral of x^2*exp(3*x) dx

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The solution

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∫01x2e3x dx\int\limits_{0}^{1} x^{2} e^{3 x}\, dx 
Integral(x^2*exp(3*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)=e3x\operatorname{dv}{\left(x \right)} = e^{3 x}.

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

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

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      ∫eu3 du\int \frac{e^{u}}{3}\, 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: eu3\frac{e^{u}}{3}

      Now substitute uu back in:

      e3x3\frac{e^{3 x}}{3}

    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)=2x3u{\left(x \right)} = \frac{2 x}{3} and let dv⁡(x)=e3x\operatorname{dv}{\left(x \right)} = e^{3 x}.

    Then du⁡(x)=23\operatorname{du}{\left(x \right)} = \frac{2}{3}.

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

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      ∫eu3 du\int \frac{e^{u}}{3}\, 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: eu3\frac{e^{u}}{3}

      Now substitute uu back in:

      e3x3\frac{e^{3 x}}{3}

    Now evaluate the sub-integral.

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

    ∫2e3x9 dx=2∫e3x dx9\int \frac{2 e^{3 x}}{9}\, dx = \frac{2 \int e^{3 x}\, dx}{9}

    1. Let u=3xu = 3 x.

      Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

      ∫eu3 du\int \frac{e^{u}}{3}\, 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: eu3\frac{e^{u}}{3}

      Now substitute uu back in:

      e3x3\frac{e^{3 x}}{3}

    So, the result is: 2e3x27\frac{2 e^{3 x}}{27}

  4. Now simplify:

    (9x2−6x+2)e3x27\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}

  5. Add the constant of integration:

    (9x2−6x+2)e3x27+constant\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}+ \mathrm{constant}

The answer is:

(9x2−6x+2)e3x27+constant\frac{\left(9 x^{2} - 6 x + 2\right) e^{3 x}}{27}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                            
 |                     3*x        3*x    2  3*x
 |  2  3*x          2*e      2*x*e      x *e   
 | x *e    dx = C + ------ - -------- + -------
 |                    27        9          3   
/
∫x2e3x dx=C+x2e3x3−2xe3x9+2e3x27\int x^{2} e^{3 x}\, dx = C + \frac{x^{2} e^{3 x}}{3} - \frac{2 x e^{3 x}}{9} + \frac{2 e^{3 x}}{27}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90040
Plot the graph f(x)
The answer [src] 
          3
  2    5*e 
- -- + ----
  27    27
−227+5e327- \frac{2}{27} + \frac{5 e^{3}}{27} 
=
= 
          3
  2    5*e 
- -- + ----
  27    27
−227+5e327- \frac{2}{27} + \frac{5 e^{3}}{27} 
-2/27 + 5*exp(3)/27
Numerical answer [src] 
3.64546980059031
3.64546980059031
The graph
Integral of x^2*exp(3*x) dx

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

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