Integral of x/(1-x) dx

The solution

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

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

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

Detail solution

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

$- x - \log{\left(x - 1 \right)}+ \mathrm{constant}$

The answer is:

$- x - \log{\left(x - 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo + pi*I

$$\infty + i \pi$$

oo + pi*I

$$\infty + i \pi$$

oo + pi*i

Numerical answer [src]

43.0909567862195

43.0909567862195

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/(1-x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2