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Identical expressions
((ln(two x))^2)/x
((ln(2x)) squared ) divide by x
((ln(two x)) squared ) divide by x
((ln(2x))2)/x
ln2x2/x
((ln(2x))²)/x
((ln(2x)) to the power of 2)/x
ln2x^2/x
((ln(2x))^2) divide by x
((ln(2x))^2)/xdx

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

The solution

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

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

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

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \log{\left(2 x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $du$ :
  
  $\int u^{2}\, du$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int u^{2}\, du = \frac{u^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(2 x \right)}^{3}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\frac{\log{\left(2 x \right)}^{2}}{x} = \frac{\log{\left(x \right)}^{2} + 2 \log{\left(2 \right)} \log{\left(x \right)} + \log{\left(2 \right)}^{2}}{x}$
2. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $du$ :
  
  $\int \frac{- \log{\left(\frac{1}{u} \right)}^{2} - 2 \log{\left(2 \right)} \log{\left(\frac{1}{u} \right)} - \log{\left(2 \right)}^{2}}{u}\, du$
  1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
    
    Then let $du = - \frac{du}{u}$ and substitute $du$ :
    
    $\int \left(u^{2} + 2 u \log{\left(2 \right)} + \log{\left(2 \right)}^{2}\right)\, du$
    1. Integrate term-by-term:
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 2 u \log{\left(2 \right)}\, du = 2 \log{\left(2 \right)} \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $u^{2} \log{\left(2 \right)}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \log{\left(2 \right)}^{2}\, du = u \log{\left(2 \right)}^{2}$
      
      The result is: $\frac{u^{3}}{3} + u^{2} \log{\left(2 \right)} + u \log{\left(2 \right)}^{2}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(\frac{1}{u} \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(\frac{1}{u} \right)}$
  Now substitute $u$ back in:
  
  $\frac{\log{\left(x \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(x \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{\log{\left(2 x \right)}^{2}}{x} = \frac{\log{\left(x \right)}^{2}}{x} + \frac{2 \log{\left(2 \right)} \log{\left(x \right)}}{x} + \frac{\log{\left(2 \right)}^{2}}{x}$
2. Integrate term-by-term:
  1. Let $u = \frac{1}{x}$ .
    
    Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int u^{2}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{3}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(x \right)}^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2 \log{\left(2 \right)} \log{\left(x \right)}}{x}\, dx = 2 \log{\left(2 \right)} \int \frac{\log{\left(x \right)}}{x}\, dx$
    1. Let $u = \frac{1}{x}$ .
      
      Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
      
      $\int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}}{u}\right)\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int u\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u\right)\, du = - \int u\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u\, du = \frac{u^{2}}{2}$
      
      So, the result is: $- \frac{u^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{2}}{2}$
      
      Now substitute $u$ back in:
      
      $\frac{\log{\left(x \right)}^{2}}{2}$
    So, the result is: $\log{\left(2 \right)} \log{\left(x \right)}^{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(2 \right)}^{2}}{x}\, dx = \log{\left(2 \right)}^{2} \int \frac{1}{x}\, dx$
    1. The integral of $\frac{1}{x}$ is $\log{\left(x \right)}$ .
    So, the result is: $\log{\left(2 \right)}^{2} \log{\left(x \right)}$
  The result is: $\frac{\log{\left(x \right)}^{3}}{3} + \log{\left(2 \right)} \log{\left(x \right)}^{2} + \log{\left(2 \right)}^{2} \log{\left(x \right)}$
Add the constant of integration:

$\frac{\log{\left(2 x \right)}^{3}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(2 x \right)}^{3}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                            
 |                             
 |    2                  3     
 | log (2*x)          log (2*x)
 | --------- dx = C + ---------
 |     x                  3    
 |                             
/

$${{\left(\log \left(2\,x\right)\right)^3}\over{3}}$$

Simplify

The answer [src]

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$${\it \%a}$$

oo

$$\infty$$

Упростить

Numerical answer [src]

27242.1350802983

27242.1350802983

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

Other calculators

How to use it?

Identical expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3