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(1-7x)e^(5x)

Integral of (1-7x)e^(5x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
  1                  
  /                  
 |                   
 |             5*x   
 |  (1 - 7*x)*e    dx
 |                   
/                    
0                    
$$\int\limits_{0}^{1} \left(1 - 7 x\right) e^{5 x}\, dx$$
Integral((1 - 7*x)*E^(5*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]
  /                                          
 |                             5*x        5*x
 |            5*x          12*e      7*x*e   
 | (1 - 7*x)*e    dx = C + ------- - --------
 |                            25        5    
/                                            
$$\int \left(1 - 7 x\right) e^{5 x}\, dx = C - \frac{7 x e^{5 x}}{5} + \frac{12 e^{5 x}}{25}$$
The graph
The answer [src]
           5
  12   23*e 
- -- - -----
  25     25 
$$- \frac{23 e^{5}}{25} - \frac{12}{25}$$
=
=
           5
  12   23*e 
- -- - -----
  25     25 
$$- \frac{23 e^{5}}{25} - \frac{12}{25}$$
Numerical answer [src]
-137.02010637437
-137.02010637437
The graph
Integral of (1-7x)e^(5x) dx

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