Integral of x/(x-2) dx

The solution

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x}{x - 2} = 1 + \frac{2}{x - 2}$
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:
  
  $\int \frac{2}{x - 2}\, dx = 2 \int \frac{1}{x - 2}\, dx$
  1. Let $u = x - 2$ .
    
    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:
    
    $\log{\left(x - 2 \right)}$
  So, the result is: $2 \log{\left(x - 2 \right)}$
The result is: $x + 2 \log{\left(x - 2 \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$x+2\,\log \left(x-2\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

1 - 2*log(2)

$$1-2\,\log 2$$

1 - 2*log(2)

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

Simplify

Numerical answer [src]

-0.386294361119891

-0.386294361119891

The graph

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

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Integral of x/(x-2) dx

Limits of integration:

The graph:

Piecewise:

The solution