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

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  1                  
  /                  
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 |             5*x   
 |  (3*x - 2)*e    dx
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/                    
0

$$\int\limits_{0}^{1} \left(3 x - 2\right) e^{5 x}\, dx$$

Integral((3*x - 1*2)*E^(5*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(3 x - 2\right) e^{5 x} = 3 x e^{5 x} - 2 e^{5 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x e^{5 x}\, dx = 3 \int x e^{5 x}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
      
      Method #2
      
      Let $u = e^{5 x}$ .
      
      Then let $du = 5 e^{5 x} dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{1}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{5}\, du = \frac{\int 1\, du}{5}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      So, the result is: $\frac{u}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{5 x}}{5}\, dx = \frac{\int e^{5 x}\, dx}{5}$
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
      
      So, the result is: $\frac{e^{5 x}}{25}$
    So, the result is: $\frac{3 x e^{5 x}}{5} - \frac{3 e^{5 x}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 e^{5 x}\right)\, dx = - 2 \int e^{5 x}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
    So, the result is: $- \frac{2 e^{5 x}}{5}$
  The result is: $\frac{3 x e^{5 x}}{5} - \frac{13 e^{5 x}}{25}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 3 x - 2$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\operatorname{du}{\left(x \right)} = 3$ .
  
  To find $v{\left(x \right)}$ :
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{e^{u}}{25}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
    Now substitute $u$ back in:
    
    $\frac{e^{5 x}}{5}$
  Now evaluate the sub-integral.
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{3 e^{5 x}}{5}\, dx = \frac{3 \int e^{5 x}\, dx}{5}$
  1. Let $u = 5 x$ .
    
    Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
    
    $\int \frac{e^{u}}{25}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
    Now substitute $u$ back in:
    
    $\frac{e^{5 x}}{5}$
  So, the result is: $\frac{3 e^{5 x}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(3 x - 2\right) e^{5 x} = 3 x e^{5 x} - 2 e^{5 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 3 x e^{5 x}\, dx = 3 \int x e^{5 x}\, dx$
    1. Use integration by parts:
      
      $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
      
      Let $u{\left(x \right)} = x$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
      
      Then $\operatorname{du}{\left(x \right)} = 1$ .
      
      To find $v{\left(x \right)}$ :
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
      
      Now evaluate the sub-integral.
    2. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{5 x}}{5}\, dx = \frac{\int e^{5 x}\, dx}{5}$
      
      Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
      
      So, the result is: $\frac{e^{5 x}}{25}$
    So, the result is: $\frac{3 x e^{5 x}}{5} - \frac{3 e^{5 x}}{25}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 2 e^{5 x}\right)\, dx = - 2 \int e^{5 x}\, dx$
    1. Let $u = 5 x$ .
      
      Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
      
      $\int \frac{e^{u}}{25}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{5}\, du = \frac{\int e^{u}\, du}{5}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{5}$
      
      Now substitute $u$ back in:
      
      $\frac{e^{5 x}}{5}$
    So, the result is: $- \frac{2 e^{5 x}}{5}$
  The result is: $\frac{3 x e^{5 x}}{5} - \frac{13 e^{5 x}}{25}$
Now simplify:

$\frac{\left(15 x - 13\right) e^{5 x}}{25}$
Add the constant of integration:

$\frac{\left(15 x - 13\right) e^{5 x}}{25}+ \mathrm{constant}$

The answer is:

$\frac{\left(15 x - 13\right) e^{5 x}}{25}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                          
 |                             5*x        5*x
 |            5*x          13*e      3*x*e   
 | (3*x - 2)*e    dx = C - ------- + --------
 |                            25        5    
/

$${{3\,\left(5\,x-1\right)\,e^{5\,x}}\over{25}}-{{2\,e^{5\,x}}\over{5 }}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

        5
13   2*e 
-- + ----
25    25

$${{2\,e^5}\over{25}}+{{13}\over{25}}$$

=

        5
13   2*e 
-- + ----
25    25

$$\frac{13}{25} + \frac{2 e^{5}}{25}$$

Simplify

Numerical answer [src]

12.3930527282061

12.3930527282061

The graph

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2

Method #3