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Derivative of:
(x-1)/(x-3)
Identical expressions
(x- one)/(x- three)
(x minus 1) divide by (x minus 3)
(x minus one) divide by (x minus three)
x-1/x-3
(x-1) divide by (x-3)
(x-1)/(x-3)dx
Similar expressions
(x-1)/(x+3)
(x-1)/((x-3)*(x-4))
(x+1)/(x-3)
(7x-1)/((x-3)^2+4)

Solving integrals/ (x-1)/(x-3)

Integral of (x-1)/(x-3) dx

The solution

You have entered [src]

  1         
  /         
 |          
 |  x - 1   
 |  ----- dx
 |  x - 3   
 |          
/           
0

\int\limits_{0}^{1} \frac{x - 1}{x - 3}\, dx

Integral((x - 1)/(x - 3), (x, 0, 1))

Detail solution

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

$x + 2 \log{\left(x - 3 \right)}+ \mathrm{constant}$

The answer is:

x + 2 \log{\left(x - 3 \right)}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                                
 |                                 
 | x - 1                           
 | ----- dx = C + x + 2*log(-3 + x)
 | x - 3                           
 |                                 
/

\int \frac{x - 1}{x - 3}\, dx = C + x + 2 \log{\left(x - 3 \right)}

Simplify

The graph

Plot the graph f(x)

The answer [src]

1 - 2*log(3) + 2*log(2)

- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)}

1 - 2*log(3) + 2*log(2)

- 2 \log{\left(3 \right)} + 1 + 2 \log{\left(2 \right)}

1 - 2*log(3) + 2*log(2)

Numerical answer [src]

0.189069783783671

0.189069783783671

The graph

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

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Identical expressions

Similar expressions

Integral of (x-1)/(x-3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2