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

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  1                  
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 |             5*x   
 |  (1 - 7*x)*e    dx
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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

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\left(1 - 7 x\right) e^{5 x} = - 7 x e^{5 x} + e^{5 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 7 x e^{5 x}\right)\, dx = - 7 \int x e^{5 x}\, dx$
    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: $- \frac{7 x e^{5 x}}{5} + \frac{7 e^{5 x}}{25}$
  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:
    
    $\frac{e^{5 x}}{5}$
  The result is: $- \frac{7 x e^{5 x}}{5} + \frac{12 e^{5 x}}{25}$
Method #2
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = 1 - 7 x$ and let $\operatorname{dv}{\left(x \right)} = e^{5 x}$ .
  
  Then $\operatorname{du}{\left(x \right)} = -7$ .
  
  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:
  
  $\int \left(- \frac{7 e^{5 x}}{5}\right)\, dx = - \frac{7 \int e^{5 x}\, dx}{5}$
  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: $- \frac{7 e^{5 x}}{25}$
Method #3
1. Rewrite the integrand:
  
  $\left(1 - 7 x\right) e^{5 x} = - 7 x e^{5 x} + e^{5 x}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- 7 x e^{5 x}\right)\, dx = - 7 \int x e^{5 x}\, dx$
    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: $- \frac{7 x e^{5 x}}{5} + \frac{7 e^{5 x}}{25}$
  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:
    
    $\frac{e^{5 x}}{5}$
  The result is: $- \frac{7 x e^{5 x}}{5} + \frac{12 e^{5 x}}{25}$
Now simplify:

$\frac{\left(12 - 35 x\right) e^{5 x}}{25}$
Add the constant of integration:

$\frac{\left(12 - 35 x\right) e^{5 x}}{25}+ \mathrm{constant}$

The answer is:

\frac{\left(12 - 35 x\right) e^{5 x}}{25}+ \mathrm{constant}

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}

Simplify

The graph

Plot the graph f(x)

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}

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Numerical answer [src]

-137.02010637437

-137.02010637437

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

Limits of integration:

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The solution

Method #1

Method #1

Method #2

Method #2

Method #3