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Integral of d{x}:
Integral of e^x/x^2
Integral of (1+x)/x
Integral of coshx
Integral of a^x*e^x
Identical expressions
(x- one)/ln(x)
(x minus 1) divide by ln(x)
(x minus one) divide by ln(x)
x-1/lnx
(x-1) divide by ln(x)
(x-1)/ln(x)dx
Similar expressions
(x+1)/ln(x)
(x-1)/(lnx)^(1/3)
(exp^(1/x)-1)/((ln(x+7))^2)
(x-1)/(ln(x))

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

The solution

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  1          
  /          
 |           
 |  x - 1    
 |  ------ dx
 |  log(x)   
 |           
/            
0

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $du$ :
  
  $\int \frac{e^{2 u} - e^{u}}{u}\, du$
  1. Let $u = \frac{1}{u}$ .
    
    Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
    
    $\int \left(- \frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u}\, du = - \int \frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u}\, du$
      
      Rewrite the integrand:
      
      $\frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u} = \frac{e^{\frac{2}{u}}}{u} - \frac{e^{\frac{1}{u}}}{u}$
      
      Integrate term-by-term:
      
      Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{e^{2 u}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{2 u}}{u}\, du = - \int \frac{e^{2 u}}{u}\, du$
      
      EiRule(a=2, b=0, context=exp(2*_u)/_u, symbol=_u)
      
      So, the result is: $- \operatorname{Ei}{\left(2 u \right)}$
      
      Now substitute $u$ back in:
      
      $- \operatorname{Ei}{\left(\frac{2}{u} \right)}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{e^{\frac{1}{u}}}{u}\right)\, du = - \int \frac{e^{\frac{1}{u}}}{u}\, du$
      
      Let $u = \frac{1}{u}$ .
      
      Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$ :
      
      $\int \left(- \frac{e^{u}}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{e^{u}}{u}\, du = - \int \frac{e^{u}}{u}\, du$
      
      EiRule(a=1, b=0, context=exp(_u)/_u, symbol=_u)
      
      So, the result is: $- \operatorname{Ei}{\left(u \right)}$
      
      Now substitute $u$ back in:
      
      $- \operatorname{Ei}{\left(\frac{1}{u} \right)}$
      
      So, the result is: $\operatorname{Ei}{\left(\frac{1}{u} \right)}$
      
      The result is: $\operatorname{Ei}{\left(\frac{1}{u} \right)} - \operatorname{Ei}{\left(\frac{2}{u} \right)}$
      
      So, the result is: $- \operatorname{Ei}{\left(\frac{1}{u} \right)} + \operatorname{Ei}{\left(\frac{2}{u} \right)}$
    Now substitute $u$ back in:
    
    $\operatorname{Ei}{\left(2 u \right)} - \operatorname{Ei}{\left(u \right)}$
  Now substitute $u$ back in:
  
  $- \operatorname{Ei}{\left(\log{\left(x \right)} \right)} + \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x - 1}{\log{\left(x \right)}} = \frac{x}{\log{\left(x \right)}} - \frac{1}{\log{\left(x \right)}}$
2. Integrate term-by-term:
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int \frac{e^{2 u}}{u}\, du$
    Now substitute $u$ back in:
    
    $\operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{1}{\log{\left(x \right)}}\right)\, dx = - \int \frac{1}{\log{\left(x \right)}}\, dx$
    So, the result is: $- \operatorname{li}{\left(x \right)}$
  The result is: $\operatorname{Ei}{\left(2 \log{\left(x \right)} \right)} - \operatorname{li}{\left(x \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The answer [src]

log(2)

$$\log{\left(2 \right)}$$

log(2)

$$\log{\left(2 \right)}$$

log(2)

Numerical answer [src]

0.693147180559945

0.693147180559945

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

Other calculators

How to use it?

Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2