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Solving integrals/ (x-1)/x(x+1)

Integral of (x-1)/x(x+1) dx

The solution

You have entered [src]

  1                 
  /                 
 |                  
 |  x - 1           
 |  -----*(x + 1) dx
 |    x             
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{x - 1}{x} \left(x + 1\right)\, dx$$

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

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-43.5904461339929

-43.5904461339929

The graph

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

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Identical expressions

Similar expressions

Integral of (x-1)/x(x+1) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2