Mister Exam

Integral of 3xe^-x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1           
  /           
 |            
 |       -x   
 |  3*x*e   dx
 |            
/             
0             
$$\int\limits_{0}^{1} 3 x e^{- x}\, dx$$
Integral(3*x/E^x, (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    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:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                
 |                                 
 |      -x             -x        -x
 | 3*x*e   dx = C - 3*e   - 3*x*e  
 |                                 
/                                  
$$3\,\left(-x-1\right)\,e^ {- x }$$
The graph
The answer [src]
       -1
3 - 6*e  
$$3\,\left(1-2\,e^ {- 1 }\right)$$
=
=
       -1
3 - 6*e  
$$3 - \frac{6}{e}$$
Numerical answer [src]
0.792723352971346
0.792723352971346
The graph
Integral of 3xe^-x dx

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