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Identical expressions
one /(six -x)
1 divide by (6 minus x)
one divide by (six minus x)
1/6-x
1 divide by (6-x)
1/(6-x)dx
Similar expressions
1/(6+x)
x^(1/6)-x^(-1/4)

Integral of 1/(6-x) dx

The solution

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  3         
  /         
 |          
 |    1     
 |  ----- dx
 |  6 - x   
 |          
/           
1

$$\int\limits_{1}^{3} \frac{1}{6 - x}\, dx$$

Integral(1/(6 - x), (x, 1, 3))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 6 - x$ .
  
  Then let $du = - dx$ and substitute $- du$ :
  
  $\int \left(- \frac{1}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \log{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \log{\left(6 - x \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{1}{6 - x} = - \frac{1}{x - 6}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x - 6}\right)\, dx = - \int \frac{1}{x - 6}\, dx$
  1. Let $u = x - 6$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x - 6 \right)}$
  So, the result is: $- \log{\left(x - 6 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{1}{6 - x} = - \frac{1}{x - 6}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{x - 6}\right)\, dx = - \int \frac{1}{x - 6}\, dx$
  1. Let $u = x - 6$ .
    
    Then let $du = dx$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    Now substitute $u$ back in:
    
    $\log{\left(x - 6 \right)}$
  So, the result is: $- \log{\left(x - 6 \right)}$
Add the constant of integration:

$- \log{\left(6 - x \right)}+ \mathrm{constant}$

The answer is:

$- \log{\left(6 - x \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                         
 |                          
 |   1                      
 | ----- dx = C - log(6 - x)
 | 6 - x                    
 |                          
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-log(3) + log(5)

$$- \log{\left(3 \right)} + \log{\left(5 \right)}$$

-log(3) + log(5)

$$- \log{\left(3 \right)} + \log{\left(5 \right)}$$

-log(3) + log(5)

Numerical answer [src]

0.510825623765991

0.510825623765991

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

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Identical expressions

Similar expressions

Integral of 1/(6-x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3