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Integral of d{x}:
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Integral of sin(log2(x))/x
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Identical expressions
sin(log2(x))/x
sinus of ( logarithm of 2(x)) divide by x
sinlog2x/x
sin(log2(x)) divide by x
sin(log2(x))/xdx
Similar expressions
sin(log2x)/x

Integral of sin(log2(x))/x dx

The solution

You have entered [src]

  1               
  /               
 |                
 |     /log(x)\   
 |  sin|------|   
 |     \log(2)/   
 |  ----------- dx
 |       x        
 |                
/                 
0

$$\int\limits_{0}^{1} \frac{\sin{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}}{x}\, dx$$

Integral(sin(log(x)/log(2))/x, (x, 0, 1))

Detail solution

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

$- \log{\left(2 \right)} \cos{\left(\frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                       
 |                                        
 |    /log(x)\                            
 | sin|------|                            
 |    \log(2)/             /log(x)\       
 | ----------- dx = C - cos|------|*log(2)
 |      x                  \log(2)/       
 |                                        
/

$$-\log 2\,\cos \left({{\log x}\over{\log 2}}\right)$$

Simplify

The answer [src]

-log(2) - <-1, 1>*log(2)

$$\int_{0}^{1}{{{\sin \left({{\log x}\over{\log 2}}\right)}\over{x}} \;dx}$$

-log(2) - <-1, 1>*log(2)

$$- \log{\left(2 \right)} \left\langle -1, 1\right\rangle - \log{\left(2 \right)}$$

Simplify

Numerical answer [src]

-0.177999323190111

-0.177999323190111

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

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Identical expressions

Similar expressions

Integral of sin(log2(x))/x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2