Mister Exam

Other calculators


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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |   2  2*x   
 |  x *e    dx
 |            
/             
0             
$$\int\limits_{0}^{1} x^{2} e^{2 x}\, dx$$
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]
  /                                        
 |                   2*x    2  2*x      2*x
 |  2  2*x          e      x *e      x*e   
 | x *e    dx = C + ---- + ------- - ------
 |                   4        2        2   
/                                          
$${{\left(2\,x^2-2\,x+1\right)\,e^{2\,x}}\over{4}}$$
The graph
The answer [src]
       2
  1   e 
- - + --
  4   4 
$${{e^2}\over{4}}-{{1}\over{4}}$$
=
=
       2
  1   e 
- - + --
  4   4 
$$- \frac{1}{4} + \frac{e^{2}}{4}$$
Numerical answer [src]
1.59726402473266
1.59726402473266
The graph
Integral of x^(2)*e^(2x) dx

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