Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of √(9-x^2)
Integral of (2x+1)^(1/2)
Integral of (2-x^4)/(1+x^2)
Integral of 2*exp(x)
Identical expressions
one /(one -(x+ one)^(one / three))
1 divide by (1 minus (x plus 1) to the power of (1 divide by 3))
one divide by (one minus (x plus one) to the power of (one divide by three))
1/(1-(x+1)(1/3))
1/1-x+11/3
1/1-x+1^1/3
1 divide by (1-(x+1)^(1 divide by 3))
1/(1-(x+1)^(1/3))dx
Similar expressions
1/(1-(x-1)^(1/3))
1/(1+(x+1)^(1/3))

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

The solution

You have entered [src]

  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.

Other calculators

How to use it?

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