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Identical expressions
one /((z^(one / four))- four)
1 divide by ((z to the power of (1 divide by 4)) minus 4)
one divide by ((z to the power of (one divide by four)) minus four)
1/((z(1/4))-4)
1/z1/4-4
1/z^1/4-4
1 divide by ((z^(1 divide by 4))-4)
Similar expressions
1/((z^(1/4))+4)

Integral of 1/((z^(1/4))-4) dx

The solution

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 81             
  /             
 |              
 |      1       
 |  --------- dz
 |  4 ___       
 |  \/ z  - 4   
 |              
/               
0

$$\int\limits_{0}^{81} \frac{1}{\sqrt[4]{z} - 4}\, dz$$

Integral(1/(z^(1/4) - 4), (z, 0, 81))

Detail solution

Let $u = \sqrt[4]{z}$ .

Then let $du = \frac{dz}{4 z^{\frac{3}{4}}}$ and substitute $4 du$ :

$\int \frac{4 u^{3}}{u - 4}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{u^{3}}{u - 4}\, du = 4 \int \frac{u^{3}}{u - 4}\, du$
  1. Rewrite the integrand:
    
    $\frac{u^{3}}{u - 4} = u^{2} + 4 u + 16 + \frac{64}{u - 4}$
  2. Integrate term-by-term:
    1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 4 u\, du = 4 \int u\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int u\, du = \frac{u^{2}}{2}$
      So, the result is: $2 u^{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 16\, du = 16 u$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{64}{u - 4}\, du = 64 \int \frac{1}{u - 4}\, du$
      1. Let $u = u - 4$ .
        
        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:
        
        $\log{\left(u - 4 \right)}$
      So, the result is: $64 \log{\left(u - 4 \right)}$
    The result is: $\frac{u^{3}}{3} + 2 u^{2} + 16 u + 64 \log{\left(u - 4 \right)}$
  So, the result is: $\frac{4 u^{3}}{3} + 8 u^{2} + 64 u + 256 \log{\left(u - 4 \right)}$
Now substitute $u$ back in:

$\frac{4 z^{\frac{3}{4}}}{3} + 64 \sqrt[4]{z} + 8 \sqrt{z} + 256 \log{\left(\sqrt[4]{z} - 4 \right)}$
Add the constant of integration:

$\frac{4 z^{\frac{3}{4}}}{3} + 64 \sqrt[4]{z} + 8 \sqrt{z} + 256 \log{\left(\sqrt[4]{z} - 4 \right)}+ \mathrm{constant}$

The answer is:

$\frac{4 z^{\frac{3}{4}}}{3} + 64 \sqrt[4]{z} + 8 \sqrt{z} + 256 \log{\left(\sqrt[4]{z} - 4 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                    
 |                                                                  3/4
 |     1                  ___      4 ___          /     4 ___\   4*z   
 | --------- dz = C + 8*\/ z  + 64*\/ z  + 256*log\-4 + \/ z / + ------
 | 4 ___                                                           3   
 | \/ z  - 4                                                           
 |                                                                     
/

$$\int \frac{1}{\sqrt[4]{z} - 4}\, dz = C + \frac{4 z^{\frac{3}{4}}}{3} + 64 \sqrt[4]{z} + 8 \sqrt{z} + 256 \log{\left(\sqrt[4]{z} - 4 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

 81                                                            
  /                                                            
 |                                                             
 |  /                                              4 ___       
 |  |  1       4      16            16             \/ z        
 |  |----- + ----- + ---- + -----------------  for ----- > 1   
 |  |4 ___     ___    3/4        /     4 ___\        4         
 |  |\/ z    \/ z    z       3/4 |     \/ z |                  
 |  |                       z   *|-1 + -----|                  
 |  |                            \       4  /                  
 |  <                                                        dz
 |  |  1       4      16           16                          
 |  |----- + ----- + ---- - ----------------     otherwise     
 |  |4 ___     ___    3/4        /    4 ___\                   
 |  |\/ z    \/ z    z       3/4 |    \/ z |                   
 |  |                       z   *|1 - -----|                   
 |  |                            \      4  /                   
 |  \                                                          
 |                                                             
/                                                              
0

$$\int\limits_{0}^{81} \begin{cases} \frac{4}{\sqrt{z}} + \frac{1}{\sqrt[4]{z}} + \frac{16}{z^{\frac{3}{4}}} + \frac{16}{z^{\frac{3}{4}} \left(\frac{\sqrt[4]{z}}{4} - 1\right)} & \text{for}\: \frac{\sqrt[4]{z}}{4} > 1 \\\frac{4}{\sqrt{z}} + \frac{1}{\sqrt[4]{z}} + \frac{16}{z^{\frac{3}{4}}} - \frac{16}{z^{\frac{3}{4}} \left(1 - \frac{\sqrt[4]{z}}{4}\right)} & \text{otherwise} \end{cases}\, dz$$

 81                                                            
  /                                                            
 |                                                             
 |  /                                              4 ___       
 |  |  1       4      16            16             \/ z        
 |  |----- + ----- + ---- + -----------------  for ----- > 1   
 |  |4 ___     ___    3/4        /     4 ___\        4         
 |  |\/ z    \/ z    z       3/4 |     \/ z |                  
 |  |                       z   *|-1 + -----|                  
 |  |                            \       4  /                  
 |  <                                                        dz
 |  |  1       4      16           16                          
 |  |----- + ----- + ---- - ----------------     otherwise     
 |  |4 ___     ___    3/4        /    4 ___\                   
 |  |\/ z    \/ z    z       3/4 |    \/ z |                   
 |  |                       z   *|1 - -----|                   
 |  |                            \      4  /                   
 |  \                                                          
 |                                                             
/                                                              
0

Integral(Piecewise((z^(-1/4) + 4/sqrt(z) + 16/z^(3/4) + 16/(z^(3/4)*(-1 + z^(1/4)/4)), z^(1/4)/4 > 1), (z^(-1/4) + 4/sqrt(z) + 16/z^(3/4) - 16/(z^(3/4)*(1 - z^(1/4)/4)), True)), (z, 0, 81))

Numerical answer [src]

-54.891356446692

-54.891356446692

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

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Identical expressions

Similar expressions

Integral of 1/((z^(1/4))-4) dx

Limits of integration:

The graph:

Piecewise:

The solution