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Identical expressions
y/(one -y)
y divide by (1 minus y)
y divide by (one minus y)
y/1-y
y divide by (1-y)
Similar expressions
y/(1+y)

Solving integrals/ y/(1-y)

Integral of y/(1-y) dx

The solution

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  1         
  /         
 |          
 |    y     
 |  ----- dy
 |  1 - y   
 |          
/           
0

$$\int\limits_{0}^{1} \frac{y}{1 - y}\, dy$$

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                              
 |                               
 |   y                           
 | ----- dy = C - y - log(-1 + y)
 | 1 - y                         
 |                               
/

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

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.

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Identical expressions

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Integral of y/(1-y) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2