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x*e^(3*x)

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1          
  /          
 |           
 |     3*x   
 |  x*e    dx
 |           
/            
0            
$$\int\limits_{0}^{1} x e^{3 x}\, dx$$
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. 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:

  3. Now simplify:

  4. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                             
 |                  3*x      3*x
 |    3*x          e      x*e   
 | x*e    dx = C - ---- + ------
 |                  9       3   
/                               
$${{\left(3\,x-1\right)\,e^{3\,x}}\over{9}}$$
The graph
The answer [src]
       3
1   2*e 
- + ----
9    9  
$${{2\,e^3}\over{9}}+{{1}\over{9}}$$
=
=
       3
1   2*e 
- + ----
9    9  
$$\frac{1}{9} + \frac{2 e^{3}}{9}$$
Numerical answer [src]
4.57456376070837
4.57456376070837
The graph
Integral of x*e^(3*x) dx

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