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

The solution

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  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |      3 _______   
 |  1 - \/ x + 1    
 |                  
/                   
0

$$\int\limits_{0}^{1} \frac{1}{1 - \sqrt[3]{x + 1}}\, dx$$

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

Detail solution

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

$- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}$
Add the constant of integration:

$- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}+ \mathrm{constant}$

The answer is:

$- \frac{3 \left(x + 1\right)^{\frac{2}{3}}}{2} - 3 \sqrt[3]{x + 1} - 3 \log{\left(\sqrt[3]{x + 1} - 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

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

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

-oo

$$-\infty$$

-oo

$$-\infty$$

-oo

Numerical answer [src]

-133.185682576228

-133.185682576228

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

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Identical expressions

Similar expressions

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

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2