Integral of x/(2-x) dx

The solution

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

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

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

Detail solution

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-1 + 2*log(2)

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

-1 + 2*log(2)

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

-1 + 2*log(2)

Numerical answer [src]

0.386294361119891

0.386294361119891

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 x/(2-x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2