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(1-6x)×e^(2x)

Integral of (1-6x)×e^(2x) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src]
  1                  
  /                  
 |                   
 |             2*x   
 |  (1 - 6*x)*e    dx
 |                   
/                    
0                    
$$\int\limits_{0}^{1} \left(1 - 6 x\right) e^{2 x}\, dx$$
Integral((1 - 6*x)*E^(2*x), (x, 0, 1))
Detail solution
  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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:

      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:

      The result is:

    Method #2

    1. 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.

    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:

    Method #3

    1. Rewrite the integrand:

    2. Integrate term-by-term:

      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. 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.

        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:

      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:

      The result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                         
 |                                          
 |            2*x             2*x        2*x
 | (1 - 6*x)*e    dx = C + 2*e    - 3*x*e   
 |                                          
/                                           
$${{e^{2\,x}}\over{2}}-{{3\,\left(2\,x-1\right)\,e^{2\,x}}\over{2}}$$
The graph
The answer [src]
      2
-2 - e 
$$-e^2-2$$
=
=
      2
-2 - e 
$$- e^{2} - 2$$
Numerical answer [src]
-9.38905609893065
-9.38905609893065
The graph
Integral of (1-6x)×e^(2x) dx

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