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Integral of d{x}:
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Identical expressions
(x- one)/(lnx)^(one / three)
(x minus 1) divide by (lnx) to the power of (1 divide by 3)
(x minus one) divide by (lnx) to the power of (one divide by three)
(x-1)/(lnx)(1/3)
x-1/lnx1/3
x-1/lnx^1/3
(x-1) divide by (lnx)^(1 divide by 3)
(x-1)/(lnx)^(1/3)dx
Similar expressions
(x+1)/(lnx)^(1/3)

Integral of (x-1)/(lnx)^(1/3) dx

The solution

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  1              
  /              
 |               
 |    x - 1      
 |  ---------- dx
 |  3 ________   
 |  \/ log(x)    
 |               
/                
0

$$\int\limits_{0}^{1} \frac{x - 1}{\sqrt[3]{\log{\left(x \right)}}}\, dx$$

Integral((x - 1)/log(x)^(1/3), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{x - 1}{\sqrt[3]{\log{\left(x \right)}}} = \frac{x}{\sqrt[3]{\log{\left(x \right)}}} - \frac{1}{\sqrt[3]{\log{\left(x \right)}}}$
Integrate term-by-term:
1. Let $u = \log{\left(x \right)}$ .
  
  Then let $du = \frac{dx}{x}$ and substitute $du$ :
  
  $\int \frac{e^{2 u}}{\sqrt[3]{u}}\, du$
  Now substitute $u$ back in:
  
  $\frac{\sqrt[3]{2} \sqrt[3]{- \log{\left(x \right)}} \Gamma\left(\frac{2}{3}, - 2 \log{\left(x \right)}\right)}{2 \sqrt[3]{\log{\left(x \right)}}}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{1}{\sqrt[3]{\log{\left(x \right)}}}\right)\, dx = - \int \frac{1}{\sqrt[3]{\log{\left(x \right)}}}\, dx$
  1. Let $u = \log{\left(x \right)}$ .
    
    Then let $du = \frac{dx}{x}$ and substitute $du$ :
    
    $\int \frac{e^{u}}{\sqrt[3]{u}}\, du$
    Now substitute $u$ back in:
    
    $\frac{\sqrt[3]{- \log{\left(x \right)}} \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)}{\sqrt[3]{\log{\left(x \right)}}}$
  So, the result is: $- \frac{\sqrt[3]{- \log{\left(x \right)}} \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)}{\sqrt[3]{\log{\left(x \right)}}}$
The result is: $\frac{\sqrt[3]{2} \sqrt[3]{- \log{\left(x \right)}} \Gamma\left(\frac{2}{3}, - 2 \log{\left(x \right)}\right)}{2 \sqrt[3]{\log{\left(x \right)}}} - \frac{\sqrt[3]{- \log{\left(x \right)}} \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)}{\sqrt[3]{\log{\left(x \right)}}}$
Now simplify:

$\frac{\sqrt[3]{- \log{\left(x \right)}} \left(\frac{\sqrt[3]{2} \Gamma\left(\frac{2}{3}, - 2 \log{\left(x \right)}\right)}{2} - \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)\right)}{\sqrt[3]{\log{\left(x \right)}}}$
Add the constant of integration:

$\frac{\sqrt[3]{- \log{\left(x \right)}} \left(\frac{\sqrt[3]{2} \Gamma\left(\frac{2}{3}, - 2 \log{\left(x \right)}\right)}{2} - \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)\right)}{\sqrt[3]{\log{\left(x \right)}}}+ \mathrm{constant}$

The answer is:

$\frac{\sqrt[3]{- \log{\left(x \right)}} \left(\frac{\sqrt[3]{2} \Gamma\left(\frac{2}{3}, - 2 \log{\left(x \right)}\right)}{2} - \Gamma\left(\frac{2}{3}, - \log{\left(x \right)}\right)\right)}{\sqrt[3]{\log{\left(x \right)}}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                             
 |                     3 _________                       3 ___ 3 _________                      
 |   x - 1             \/ -log(x) *Gamma(2/3, -log(x))   \/ 2 *\/ -log(x) *Gamma(2/3, -2*log(x))
 | ---------- dx = C - ------------------------------- + ---------------------------------------
 | 3 ________                     3 ________                             3 ________             
 | \/ log(x)                      \/ log(x)                            2*\/ log(x)              
 |                                                                                              
/

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

Simplify

The answer [src]

  1              
  /              
 |               
 |    -1 + x     
 |  ---------- dx
 |  3 ________   
 |  \/ log(x)    
 |               
/                
0

$$\int\limits_{0}^{1} \frac{x - 1}{\sqrt[3]{\log{\left(x \right)}}}\, dx$$

  1              
  /              
 |               
 |    -1 + x     
 |  ---------- dx
 |  3 ________   
 |  \/ log(x)    
 |               
/                
0

$$\int\limits_{0}^{1} \frac{x - 1}{\sqrt[3]{\log{\left(x \right)}}}\, dx$$

Integral((-1 + x)/log(x)^(1/3), (x, 0, 1))

Numerical answer [src]

(-0.250538545732302 + 0.433945490462766j)

(-0.250538545732302 + 0.433945490462766j)

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

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Identical expressions

Similar expressions

Integral of (x-1)/(lnx)^(1/3) dx

Limits of integration:

The graph:

Piecewise:

The solution