Integral of x/(x-1) dx

The solution

You have entered [src]

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

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

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

Detail solution

Rewrite the integrand:

$\frac{x}{x - 1} = 1 + \frac{1}{x - 1}$
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 $du$ :
  
  $\int \frac{1}{u}\, du$
  1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
  Now substitute $u$ back in:
  
  $\log{\left(x - 1 \right)}$
The result is: $x + \log{\left(x - 1 \right)}$
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)
 | x - 1                         
 |                               
/

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

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/(x-1) dx

Limits of integration:

The graph:

Piecewise:

The solution