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Resolución de integrales/ x^(1/4)/ x^(1/2)/ x^(1/2)/(1-x^(1/4))

Integral de x^(1/2)/(1-x^(1/4)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1             
  /             
 |              
 |      ___     
 |    \/ x      
 |  --------- dx
 |      4 ___   
 |  1 - \/ x    
 |              
/               
0
∫01x1−x4 dx\int\limits_{0}^{1} \frac{\sqrt{x}}{1 - \sqrt[4]{x}}\, dx 
Integral(sqrt(x)/(1 - x^(1/4)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                              
 |                                                                               
 |     ___                                                           3/4      5/4
 |   \/ x                   4 ___        /     4 ___\       ___   4*x      4*x   
 | --------- dx = C - x - 4*\/ x  - 4*log\-1 + \/ x / - 2*\/ x  - ------ - ------
 |     4 ___                                                        3        5   
 | 1 - \/ x                                                                      
 |                                                                               
/
∫x1−x4 dx=C−4x545−4x343−4x4−2x−x−4log⁡(x4−1)\int \frac{\sqrt{x}}{1 - \sqrt[4]{x}}\, dx = C - \frac{4 x^{\frac{5}{4}}}{5} - \frac{4 x^{\frac{3}{4}}}{3} - 4 \sqrt[4]{x} - 2 \sqrt{x} - x - 4 \log{\left(\sqrt[4]{x} - 1 \right)}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.90050000
Trazar el gráfico f(x)
Respuesta [src] 
  1                                                                         
  /                                                                         
 |                                                                          
 |  /      1       1       1     4 ___           1              4 ___       
 |  |-1 - ---- - ----- - ----- - \/ x  - -----------------  for \/ x  > 1   
 |  |      3/4     ___   4 ___            3/4 /     4 ___\                  
 |  |     x      \/ x    \/ x            x   *\-1 + \/ x /                  
 |  <                                                                     dx
 |  |      1       1       1     4 ___          1                           
 |  |-1 - ---- - ----- - ----- - \/ x  + ----------------     otherwise     
 |  |      3/4     ___   4 ___            3/4 /    4 ___\                   
 |  \     x      \/ x    \/ x            x   *\1 - \/ x /                   
 |                                                                          
/                                                                           
0
∫01{−x4−1−1x−1x4−1x34−1x34(x4−1)for x4>1−x4−1−1x−1x4−1x34+1x34(1−x4)otherwise dx\int\limits_{0}^{1} \begin{cases} - \sqrt[4]{x} - 1 - \frac{1}{\sqrt{x}} - \frac{1}{\sqrt[4]{x}} - \frac{1}{x^{\frac{3}{4}}} - \frac{1}{x^{\frac{3}{4}} \left(\sqrt[4]{x} - 1\right)} & \text{for}\: \sqrt[4]{x} > 1 \\- \sqrt[4]{x} - 1 - \frac{1}{\sqrt{x}} - \frac{1}{\sqrt[4]{x}} - \frac{1}{x^{\frac{3}{4}}} + \frac{1}{x^{\frac{3}{4}} \left(1 - \sqrt[4]{x}\right)} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                         
  /                                                                         
 |                                                                          
 |  /      1       1       1     4 ___           1              4 ___       
 |  |-1 - ---- - ----- - ----- - \/ x  - -----------------  for \/ x  > 1   
 |  |      3/4     ___   4 ___            3/4 /     4 ___\                  
 |  |     x      \/ x    \/ x            x   *\-1 + \/ x /                  
 |  <                                                                     dx
 |  |      1       1       1     4 ___          1                           
 |  |-1 - ---- - ----- - ----- - \/ x  + ----------------     otherwise     
 |  |      3/4     ___   4 ___            3/4 /    4 ___\                   
 |  \     x      \/ x    \/ x            x   *\1 - \/ x /                   
 |                                                                          
/                                                                           
0
∫01{−x4−1−1x−1x4−1x34−1x34(x4−1)for x4>1−x4−1−1x−1x4−1x34+1x34(1−x4)otherwise dx\int\limits_{0}^{1} \begin{cases} - \sqrt[4]{x} - 1 - \frac{1}{\sqrt{x}} - \frac{1}{\sqrt[4]{x}} - \frac{1}{x^{\frac{3}{4}}} - \frac{1}{x^{\frac{3}{4}} \left(\sqrt[4]{x} - 1\right)} & \text{for}\: \sqrt[4]{x} > 1 \\- \sqrt[4]{x} - 1 - \frac{1}{\sqrt{x}} - \frac{1}{\sqrt[4]{x}} - \frac{1}{x^{\frac{3}{4}}} + \frac{1}{x^{\frac{3}{4}} \left(1 - \sqrt[4]{x}\right)} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-1 - 1/x^(3/4) - 1/sqrt(x) - 1/x^(1/4) - x^(1/4) - 1/(x^(3/4)*(-1 + x^(1/4))), x^(1/4) > 1), (-1 - 1/x^(3/4) - 1/sqrt(x) - 1/x^(1/4) - x^(1/4) + 1/(x^(3/4)*(1 - x^(1/4))), True)), (x, 0, 1))
Respuesta numérica [src] 
172.767574444326
172.767574444326

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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