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

The solution

You have entered [src]

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

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

Integral((1 + log(x - 1))/(x - 1), (x, -1, 1))

Detail solution

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

$\frac{\left(\log{\left(x - 1 \right)} + 1\right)^{2}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

                    2       
     (pi*I + log(2))        
oo - ---------------- - pi*I
            2

\infty - i \pi - \frac{\left(\log{\left(2 \right)} + i \pi\right)^{2}}{2}

                    2       
     (pi*I + log(2))        
oo - ---------------- - pi*I
            2

\infty - i \pi - \frac{\left(\log{\left(2 \right)} + i \pi\right)^{2}}{2}

oo - (pi*i + log(2))^2/2 - pi*i

Numerical answer [src]

(897.312398743146 - 138.514234126527j)

(897.312398743146 - 138.514234126527j)

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2