Integral of sinlnx/x dx

The solution

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  E               
  /               
 |                
 |  sin(log(x))   
 |  ----------- dx
 |       x        
 |                
/                 
0

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

Integral(sin(log(x))/x, (x, 0, E))

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                
 |                                 
 | sin(log(x))                     
 | ----------- dx = C - cos(log(x))
 |      x                          
 |                                 
/

$$\int \frac{\sin{\left(\log{\left(x \right)} \right)}}{x}\, dx = C - \cos{\left(\log{\left(x \right)} \right)}$$

Simplify

The answer [src]

<-1 - cos(1), 1 - cos(1)>

$$\left\langle -1 - \cos{\left(1 \right)}, 1 - \cos{\left(1 \right)}\right\rangle$$

<-1 - cos(1), 1 - cos(1)>

$$\left\langle -1 - \cos{\left(1 \right)}, 1 - \cos{\left(1 \right)}\right\rangle$$

AccumBounds(-1 - cos(1), 1 - cos(1))

Numerical answer [src]

0.100305548972097

0.100305548972097

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

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Identical expressions

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Integral of sinlnx/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2