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Identical expressions
(x- six)/(x- seven)^ seven
(x minus 6) divide by (x minus 7) to the power of 7
(x minus six) divide by (x minus seven) to the power of seven
(x-6)/(x-7)7
x-6/x-77
(x-6)/(x-7)⁷
x-6/x-7^7
(x-6) divide by (x-7)^7
(x-6)/(x-7)^7dx
Similar expressions
(x+6)/(x-7)^7
(x-6)/(x+7)^7

Integral of (x-6)/(x-7)^7 dx

The solution

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  1            
  /            
 |             
 |   x - 6     
 |  -------- dx
 |         7   
 |  (x - 7)    
 |             
/              
0

$$\int\limits_{0}^{1} \frac{x - 6}{\left(x - 7\right)^{7}}\, dx$$

Integral((x - 6)/(x - 7)^7, (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x - 6}{\left(x - 7\right)^{7}} = \frac{1}{\left(x - 7\right)^{6}} + \frac{1}{\left(x - 7\right)^{7}}$
2. Integrate term-by-term:
  1. Let $u = x - 7$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{6}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{5 \left(x - 7\right)^{5}}$
  1. Let $u = x - 7$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{7}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{7}}\, du = - \frac{1}{6 u^{6}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{6 \left(x - 7\right)^{6}}$
  The result is: $- \frac{1}{5 \left(x - 7\right)^{5}} - \frac{1}{6 \left(x - 7\right)^{6}}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x - 6}{\left(x - 7\right)^{7}} = \frac{x - 6}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543}$
2. Rewrite the integrand:
  
  $\frac{x - 6}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543} = \frac{1}{\left(x - 7\right)^{6}} + \frac{1}{\left(x - 7\right)^{7}}$
3. Integrate term-by-term:
  1. Let $u = x - 7$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{6}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{5 \left(x - 7\right)^{5}}$
  1. Let $u = x - 7$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u^{7}}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{7}}\, du = - \frac{1}{6 u^{6}}$
    Now substitute $u$ back in:
    
    $- \frac{1}{6 \left(x - 7\right)^{6}}$
  The result is: $- \frac{1}{5 \left(x - 7\right)^{5}} - \frac{1}{6 \left(x - 7\right)^{6}}$
Method #3
1. Rewrite the integrand:
  
  $\frac{x - 6}{\left(x - 7\right)^{7}} = \frac{x}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543} - \frac{6}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543} = \frac{1}{\left(x - 7\right)^{6}} + \frac{7}{\left(x - 7\right)^{7}}$
  2. Integrate term-by-term:
    1. Let $u = x - 7$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u^{6}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{6}}\, du = - \frac{1}{5 u^{5}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{5 \left(x - 7\right)^{5}}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{7}{\left(x - 7\right)^{7}}\, dx = 7 \int \frac{1}{\left(x - 7\right)^{7}}\, dx$
      
      Let $u = x - 7$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u^{7}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{7}}\, du = - \frac{1}{6 u^{6}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{6 \left(x - 7\right)^{6}}$
      
      So, the result is: $- \frac{7}{6 \left(x - 7\right)^{6}}$
    The result is: $- \frac{1}{5 \left(x - 7\right)^{5}} - \frac{7}{6 \left(x - 7\right)^{6}}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{6}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543}\right)\, dx = - 6 \int \frac{1}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543}\, dx$
    1. Rewrite the integrand:
      
      $\frac{1}{x^{7} - 49 x^{6} + 1029 x^{5} - 12005 x^{4} + 84035 x^{3} - 352947 x^{2} + 823543 x - 823543} = \frac{1}{\left(x - 7\right)^{7}}$
    2. Let $u = x - 7$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u^{7}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{7}}\, du = - \frac{1}{6 u^{6}}$
      
      Now substitute $u$ back in:
      
      $- \frac{1}{6 \left(x - 7\right)^{6}}$
    So, the result is: $\frac{1}{\left(x - 7\right)^{6}}$
  The result is: $- \frac{1}{5 \left(x - 7\right)^{5}} - \frac{1}{6 \left(x - 7\right)^{6}}$
Now simplify:

$\frac{37 - 6 x}{30 \left(x - 7\right)^{6}}$
Add the constant of integration:

$\frac{37 - 6 x}{30 \left(x - 7\right)^{6}}+ \mathrm{constant}$

The answer is:

$\frac{37 - 6 x}{30 \left(x - 7\right)^{6}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                            
 |  x - 6                 1             1     
 | -------- dx = C - ----------- - -----------
 |        7                    5             6
 | (x - 7)           5*(-7 + x)    6*(-7 + x) 
 |                                            
/

$$\int \frac{x - 6}{\left(x - 7\right)^{7}}\, dx = C - \frac{1}{5 \left(x - 7\right)^{5}} - \frac{1}{6 \left(x - 7\right)^{6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  1920847   
------------
164670952320

$$\frac{1920847}{164670952320}$$

  1920847   
------------
164670952320

$$\frac{1920847}{164670952320}$$

1920847/164670952320

Numerical answer [src]

1.16647591632754e-5

1.16647591632754e-5

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

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Identical expressions

Similar expressions

Integral of (x-6)/(x-7)^7 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3