Integral of log2(x) dx

The solution

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

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

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

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}\, dx = \frac{\int \log{\left(x \right)}\, dx}{\log{\left(2 \right)}}$
1. Use integration by parts:
  
  $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$
  
  Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .
  
  Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$ .
  
  To find $v{\left(x \right)}$ :
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  Now evaluate the sub-integral.
2. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
So, the result is: $\frac{x \log{\left(x \right)} - x}{\log{\left(2 \right)}}$
Now simplify:

$\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}$
Add the constant of integration:

$\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer is:

$\frac{x \left(\log{\left(x \right)} - 1\right)}{\log{\left(2 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

$${{x\,\log x-x}\over{\log 2}}$$

Simplify

The answer [src]

 -1   
------
log(2)

$$-{{1}\over{\log 2}}$$

 -1   
------
log(2)

$$- \frac{1}{\log{\left(2 \right)}}$$

Simplify

Numerical answer [src]

-1.44269504088896

-1.44269504088896

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

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Integral of log2(x) dx

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The solution