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Identical expressions
cot(lnx)/x
cotangent of (lnx) divide by x
cotlnx/x
cot(lnx) divide by x
cot(lnx)/xdx

Integral of cot(lnx)/x dx

The solution

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

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

Simplify

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 $du$ :
  
  $\int \cot{\left(u \right)}\, du$
  1. Rewrite the integrand:
    
    $\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}$
  2. Let $u = \sin{\left(u \right)}$ .
    
    Then let $du = \cos{\left(u \right)} du$ 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:
    
    $\log{\left(\sin{\left(u \right)} \right)}$
  Now substitute $u$ back in:
  
  $\log{\left(\sin{\left(\log{\left(x \right)} \right)} \right)}$
Method #2
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \frac{\cot{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cot{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\right)\, du = - \int \frac{\cot{\left(\log{\left(\frac{1}{u} \right)} \right)}}{u}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \cot{\left(u \right)}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \cot{\left(u \right)}\right)\, du = - \int \cot{\left(u \right)}\, du$
      
      Rewrite the integrand:
      
      $\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}$
      
      Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(u \right)} \right)}$
      
      So, the result is: $- \log{\left(\sin{\left(u \right)} \right)}$
      
      Now substitute $u$ back in:
      
      $- \log{\left(\sin{\left(\log{\left(\frac{1}{u} \right)} \right)} \right)}$
    So, the result is: $\log{\left(\sin{\left(\log{\left(\frac{1}{u} \right)} \right)} \right)}$
  Now substitute $u$ back in:
  
  $\log{\left(\sin{\left(\log{\left(x \right)} \right)} \right)}$
Add the constant of integration:

$\log{\left(\sin{\left(\log{\left(x \right)} \right)} \right)}+ \mathrm{constant}$

The answer is:

$\log{\left(\sin{\left(\log{\left(x \right)} \right)} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                     
 |                                      
 | cot(log(x))                          
 | ----------- dx = C + log(sin(log(x)))
 |      x                               
 |                                      
/

$$\log \sin \log x$$

Simplify

Numerical answer [src]

-305.933349509058

-305.933349509058

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

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How to use it?

Identical expressions

Integral of cot(lnx)/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2