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1/x*(1/(ln(x))^2)dx

Integral of 1/x*(1/(ln(x))^2) dx

The solution

You have entered [src]

 oo             
  /             
 |              
 |      1       
 |  --------- dx
 |       2      
 |  x*log (x)   
 |              
/               
 2              
e

$$\int\limits_{e^{2}}^{\infty} \frac{1}{x \log{\left(x \right)}^{2}}\, dx$$

Integral(1/(x*log(x)^2), (x, exp(2), oo))

Detail solution

Let $u = \frac{1}{x}$ .

Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :

$\int \left(- \frac{1}{u \log{\left(\frac{1}{u} \right)}^{2}}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u \log{\left(\frac{1}{u} \right)}^{2}}\, du = - \int \frac{1}{u \log{\left(\frac{1}{u} \right)}^{2}}\, du$
  1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
    
    Then let $du = - \frac{du}{u}$ and substitute $- du$ :
    
    $\int \left(- \frac{1}{u^{2}}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u^{2}}\, du = - \int \frac{1}{u^{2}}\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      So, the result is: $\frac{1}{u}$
    Now substitute $u$ back in:
    
    $\frac{1}{\log{\left(\frac{1}{u} \right)}}$
  So, the result is: $- \frac{1}{\log{\left(\frac{1}{u} \right)}}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

1/2

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

1/2

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

1/2

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

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Identical expressions

Integral of 1/x*(1/(ln(x))^2) dx

Limits of integration:

The graph:

Piecewise:

The solution