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Identical expressions
ln(ln(x))/(x*ln(x))
ln(ln(x)) divide by (x multiply by ln(x))
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ln(ln(x)) divide by (x*ln(x))
ln(ln(x))/(x*ln(x))dx
Similar expressions
ln(lnx)/xlnx

Integral of ln(ln(x))/(x*ln(x)) dx

The solution

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  E               
 e                
  /               
 |                
 |  log(log(x))   
 |  ----------- dx
 |    x*log(x)    
 |                
/                 
E

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

Integral(log(log(x))/((x*log(x))), (x, E, exp(E)))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(\log{\left(x \right)} \right)}$ .
  
  Then let $du = \frac{dx}{x \log{\left(x \right)}}$ 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:
  
  $\frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{2}$
Method #2
1. Let $u = \log{\left(x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ 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 $- du$ :
    
    $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du$
    1. 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:
  
  $\frac{\log{\left(\log{\left(x \right)} \right)}^{2}}{2}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

$$\frac{1}{2}$$

1/2

$$\frac{1}{2}$$

1/2

Numerical answer [src]

0.5

0.5

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

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Identical expressions

Similar expressions

Integral of ln(ln(x))/(x*ln(x)) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2