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Identical expressions
dx/(x(loge(x))^ one / three)
dx divide by (x( logarithm of e(x)) to the power of 1 divide by 3)
dx divide by (x( logarithm of e(x)) to the power of one divide by three)
dx/(x(loge(x))1/3)
dx/xlogex1/3
dx/xlogex^1/3
dx divide by (x(loge(x))^1 divide by 3)

Integral of dx/(x(loge(x))^1/3) dx

The solution

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  E                    
  /                    
 |                     
 |         1           
 |  ---------------- dx
 |         _________   
 |        /  log(x)    
 |  x*   /  -------    
 |    3 /      / 1\    
 |    \/    log\e /    
 |                     
/                      
1

$$\int\limits_{1}^{e} \frac{1}{x \sqrt[3]{\frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}}}\, dx$$

Integral(1/(x*(log(x)/log(exp(1)))^(1/3)), (x, 1, E))

Detail solution

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                     
 |                                2/3   
 |        1                  3*log   (x)
 | ---------------- dx = C + -----------
 |        _________               2     
 |       /  log(x)                      
 | x*   /  -------                      
 |   3 /      / 1\                      
 |   \/    log\e /                      
 |                                      
/

$$\int \frac{1}{x \sqrt[3]{\frac{\log{\left(x \right)}}{\log{\left(e^{1} \right)}}}}\, dx = C + \frac{3 \log{\left(x \right)}^{\frac{2}{3}}}{2}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

3/2

$$\frac{3}{2}$$

3/2

$$\frac{3}{2}$$

3/2

Numerical answer [src]

1.49999999999955

1.49999999999955

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

Other calculators

How to use it?

Identical expressions

Integral of dx/(x(loge(x))^1/3) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #1

Method #2

Method #2