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(x-3)e^-x

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

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  3               
  /               
 |                
 |           -x   
 |  (x - 3)*E   dx
 |                
/                 
0                 
$$\int\limits_{0}^{3} e^{- x} \left(x - 3\right)\, dx$$
Integral((x - 3)*E^(-x), (x, 0, 3))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Let .

      Then let and substitute :

      1. Integrate term-by-term:

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          1. The integral of the exponential function is itself.

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

        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:

        The result is:

      Now substitute back in:

    Method #2

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      1. Let .

        Then let and substitute :

        1. Use integration by parts:

          Let and let .

          Then .

          To find :

          1. The integral of the exponential function is itself.

          Now evaluate the sub-integral.

        2. The integral of the exponential function is itself.

        Now substitute back in:

      1. 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:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                  
 |                                   
 |          -x             -x      -x
 | (x - 3)*E   dx = C + 2*e   - x*e  
 |                                   
/                                    
$$\int e^{- x} \left(x - 3\right)\, dx = C - x e^{- x} + 2 e^{- x}$$
The graph
The answer [src]
      -3
-2 - e  
$$-2 - e^{-3}$$
=
=
      -3
-2 - e  
$$-2 - e^{-3}$$
-2 - exp(-3)
Numerical answer [src]
-2.04978706836786
-2.04978706836786
The graph
Integral of (x-3)e^-x dx

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