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

The solution

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

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

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

Detail solution

Rewrite the integrand:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-oo - 2*pi*I

$$-\infty - 2 i \pi$$

-oo - 2*pi*I

$$-\infty - 2 i \pi$$

-oo - 2*pi*i

Numerical answer [src]

-86.181913572439

-86.181913572439

The graph

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

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

Limits of integration:

The graph:

Piecewise:

The solution