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x*e^(5x)

Integral of x*e^(5x) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  0          
  /          
 |           
 |     5*x   
 |  x*e    dx
 |           
/            
3            
$$\int\limits_{3}^{0} x e^{5 x}\, dx$$
Integral(x*E^(5*x), (x, 3, 0))
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]
  /                             
 |                  5*x      5*x
 |    5*x          e      x*e   
 | x*e    dx = C - ---- + ------
 |                  25      5   
/                               
$$\int x e^{5 x}\, dx = C + \frac{x e^{5 x}}{5} - \frac{e^{5 x}}{25}$$
The graph
The answer [src]
           15
  1    14*e  
- -- - ------
  25     25  
$$- \frac{14 e^{15}}{25} - \frac{1}{25}$$
=
=
           15
  1    14*e  
- -- - ------
  25     25  
$$- \frac{14 e^{15}}{25} - \frac{1}{25}$$
Numerical answer [src]
-1830649.76858438
-1830649.76858438
The graph
Integral of x*e^(5x) dx

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