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Integral of d{x}:
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Integral of x^2dx
Integral of sin²(x)
Integral of sec^2
Identical expressions
dx/xlog^ two (x)
dx divide by x logarithm of squared (x)
dx divide by x logarithm of to the power of two (x)
dx/xlog2(x)
dx/xlog2x
dx/xlog²(x)
dx/xlog to the power of 2(x)
dx/xlog^2x
dx divide by xlog^2(x)

Integral of dx/xlog^2(x) dx

The solution

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 10           
  /           
 |            
 |     2      
 |  log (x)   
 |  ------- dx
 |     x      
 |            
/             
0

$$\int\limits_{0}^{10} \frac{\log{\left(x \right)}^{2}}{x}\, dx$$

Integral(log(x)^2/x, (x, 0, 10))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \frac{1}{x}$ .
  
  Then let $du = - \frac{dx}{x^{2}}$ and substitute $- du$ :
  
  $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{2}}{u}\, du$
    1. Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u^{2}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{2}\, du = - \int u^{2}\, du$
      
      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:
  
  $\frac{\log{\left(x \right)}^{3}}{3}$
Method #2
1. Let $u = \log{\left(x \right)}$ .
  
  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:
  
  $\frac{\log{\left(x \right)}^{3}}{3}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

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

$$\int \frac{\log{\left(x \right)}^{2}}{x}\, dx = C + \frac{\log{\left(x \right)}^{3}}{3}$$

Simplify

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

Numerical answer [src]

24326.4930007318

24326.4930007318

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

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Identical expressions

Integral of dx/xlog^2(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2