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(3*x-1)/(x-2)
Identical expressions
(three *x- one)/(x- two)
(3 multiply by x minus 1) divide by (x minus 2)
(three multiply by x minus one) divide by (x minus two)
(3x-1)/(x-2)
3x-1/x-2
(3*x-1) divide by (x-2)
(3*x-1)/(x-2)dx
Similar expressions
(3*x+1)/(x-2)
(3*x-1)/(x+2)

Solving integrals/ (3*x-1)/(x-2)

Integral of (3*x-1)/(x-2) dx

The solution

You have entered [src]

  1           
  /           
 |            
 |  3*x - 1   
 |  ------- dx
 |   x - 2    
 |            
/             
0

$$\int\limits_{0}^{1} \frac{3 x - 1}{x - 2}\, dx$$

Integral((3*x - 1)/(x - 2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 3 x$ .
  
  Then let $du = 3 dx$ and substitute $du$ :
  
  $\int \frac{u - 1}{u - 6}\, du$
  1. Let $u = u - 6$ .
    
    Then let $du = du$ and substitute $du$ :
    
    $\int \frac{u + 5}{u}\, du$
    1. Rewrite the integrand:
      
      $\frac{u + 5}{u} = 1 + \frac{5}{u}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{5}{u}\, du = 5 \int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $5 \log{\left(u \right)}$
      
      The result is: $u + 5 \log{\left(u \right)}$
    Now substitute $u$ back in:
    
    $u + 5 \log{\left(u - 6 \right)} - 6$
  Now substitute $u$ back in:
  
  $3 x + 5 \log{\left(3 x - 6 \right)} - 6$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 x - 1}{x - 2} = 3 + \frac{5}{x - 2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 3\, dx = 3 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{5}{x - 2}\, dx = 5 \int \frac{1}{x - 2}\, dx$
    1. Let $u = x - 2$ .
      
      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:
      
      $\log{\left(x - 2 \right)}$
    So, the result is: $5 \log{\left(x - 2 \right)}$
  The result is: $3 x + 5 \log{\left(x - 2 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{3 x - 1}{x - 2} = \frac{3 x}{x - 2} - \frac{1}{x - 2}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x}{x - 2}\, dx = 3 \int \frac{x}{x - 2}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x - 2} = 1 + \frac{2}{x - 2}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{2}{x - 2}\, dx = 2 \int \frac{1}{x - 2}\, dx$
      
      Let $u = x - 2$ .
      
      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:
      
      $\log{\left(x - 2 \right)}$
      
      So, the result is: $2 \log{\left(x - 2 \right)}$
      
      The result is: $x + 2 \log{\left(x - 2 \right)}$
    So, the result is: $3 x + 6 \log{\left(x - 2 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{x - 2}\right)\, dx = - \int \frac{1}{x - 2}\, dx$
    1. Let $u = x - 2$ .
      
      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:
      
      $\log{\left(x - 2 \right)}$
    So, the result is: $- \log{\left(x - 2 \right)}$
  The result is: $3 x + 6 \log{\left(x - 2 \right)} - \log{\left(x - 2 \right)}$
Add the constant of integration:

$3 x + 5 \log{\left(3 x - 6 \right)} - 6+ \mathrm{constant}$

The answer is:

$3 x + 5 \log{\left(3 x - 6 \right)} - 6+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                           
 |                                            
 | 3*x - 1                                    
 | ------- dx = -6 + C + 3*x + 5*log(-6 + 3*x)
 |  x - 2                                     
 |                                            
/

$$\int \frac{3 x - 1}{x - 2}\, dx = C + 3 x + 5 \log{\left(3 x - 6 \right)} - 6$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3 - 5*log(2)

$$3 - 5 \log{\left(2 \right)}$$

3 - 5*log(2)

$$3 - 5 \log{\left(2 \right)}$$

3 - 5*log(2)

Numerical answer [src]

-0.465735902799727

-0.465735902799727

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (3*x-1)/(x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3