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(x^2)*e^(3x)

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

Limits of integration:

from to
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The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |   2  3*x   
 |  x *e    dx
 |            
/             
0             
$$\int\limits_{0}^{1} x^{2} e^{3 x}\, dx$$
Detail solution
  1. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

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

          1. The integral of the exponential function is itself.

          So, the result is:

        Now substitute back in:

      Method #2

      1. Let .

        Then let and substitute :

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

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

          So, the result is:

        Now substitute back in:

    Now evaluate the sub-integral.

  2. Use integration by parts:

    Let and let .

    Then .

    To find :

    1. Let .

      Then let and substitute :

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

        1. The integral of the exponential function is itself.

        So, the result is:

      Now substitute back in:

    Now evaluate the sub-integral.

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

    1. Let .

      Then let and substitute :

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

        1. The integral of the exponential function is itself.

        So, the result is:

      Now substitute back in:

    So, the result is:

  4. Now simplify:

  5. Add the constant of integration:


The answer is:

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   
/                                              
$${{\left(9\,x^2-6\,x+2\right)\,e^{3\,x}}\over{27}}$$
The graph
The answer [src]
          3
  2    5*e 
- -- + ----
  27    27 
$${{5\,e^3}\over{27}}-{{2}\over{27}}$$
=
=
          3
  2    5*e 
- -- + ----
  27    27 
$$- \frac{2}{27} + \frac{5 e^{3}}{27}$$
Numerical answer [src]
3.64546980059031
3.64546980059031
The graph
Integral of (x^2)*e^(3x) dx

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