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Identical expressions
dx/x(lnx)^ three / two
dx divide by x(lnx) cubed divide by 2
dx divide by x(lnx) to the power of three divide by two
dx/x(lnx)3/2
dx/xlnx3/2
dx/x(lnx)³/2
dx/x(lnx) to the power of 3/2
dx/xlnx^3/2
dx divide by x(lnx)^3 divide by 2

Integral of dx/x(lnx)^3/2 dx

The solution

You have entered [src]

 oo             
  /             
 |              
 |  /   3   \   
 |  |log (x)|   
 |  |-------|   
 |  \   x   /   
 |  --------- dx
 |      2       
 |              
/               
E

$$\int\limits_{e}^{\infty} \frac{\frac{1}{x} \log{\left(x \right)}^{3}}{2}\, dx$$

Integral((log(x)^3/x)/2, (x, E, oo))

Detail solution

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

$\int \frac{\frac{1}{x} \log{\left(x \right)}^{3}}{2}\, dx = \frac{\int \frac{\log{\left(x \right)}^{3}}{x}\, dx}{2}$
1. 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 $- du$ :
    
    $\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du$
      
      Let $u = \log{\left(\frac{1}{u} \right)}$ .
      
      Then let $du = - \frac{du}{u}$ and substitute $- du$ :
      
      $\int \left(- u^{3}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int u^{3}\, du = - \int u^{3}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
      
      So, the result is: $- \frac{u^{4}}{4}$
      
      Now substitute $u$ back in:
      
      $- \frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}$
      
      So, the result is: $\frac{\log{\left(\frac{1}{u} \right)}^{4}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(x \right)}^{4}}{4}$
  Method #2
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int u^{3}\, du$
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{3}\, du = \frac{u^{4}}{4}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(x \right)}^{4}}{4}$
So, the result is: $\frac{\log{\left(x \right)}^{4}}{8}$
Add the constant of integration:

$\frac{\log{\left(x \right)}^{4}}{8}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(x \right)}^{4}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                          
 |                           
 | /   3   \                 
 | |log (x)|                 
 | |-------|             4   
 | \   x   /          log (x)
 | --------- dx = C + -------
 |     2                 8   
 |                           
/

$$\int \frac{\frac{1}{x} \log{\left(x \right)}^{3}}{2}\, dx = C + \frac{\log{\left(x \right)}^{4}}{8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

oo

$$\infty$$

oo

$$\infty$$

oo

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

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How to use it?

Identical expressions

Integral of dx/x(lnx)^3/2 dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2