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(2-3x)e^(4x-5)

Integral of (2-3x)e^(4x-5) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src]
 5/8                     
  /                      
 |                       
 |             4*x - 5   
 |  (2 - 3*x)*e        dx
 |                       
/                        
-oo                      
$$\int\limits_{-\infty}^{\frac{5}{8}} \left(- 3 x + 2\right) e^{4 x - 5}\, dx$$
Integral((2 - 3*x)*E^(4*x - 1*5), (x, -oo, 5/8))
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. 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:

    Method #2

    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. 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]
  /                                                              
 |                              -5  4*x     /   4*x      4*x\    
 |            4*x - 5          e  *e        |  e      x*e   |  -5
 | (2 - 3*x)*e        dx = C + -------- - 3*|- ---- + ------|*e  
 |                                2         \   16      4   /    
/                                                                
$${{e^{4\,x-5}}\over{2}}-{{3\,\left(4\,x-1\right)\,e^{4\,x-5}}\over{ 16}}$$
The graph
The answer [src]
   -5/2
7*e    
-------
   32  
$${{7\,e^ {- {{5}\over{2}} }}\over{32}}-{{\left(12\,{\it oo}+11 \right)\,e^{-4\,{\it oo}-5}}\over{16}}$$
=
=
   -5/2
7*e    
-------
   32  
$$\frac{7}{32 e^{\frac{5}{2}}}$$
The graph
Integral of (2-3x)e^(4x-5) dx

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