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Resolución de integrales/ x^(-2/3)/ sqrt(1-x^(-2/3))

Integral de sqrt(1-x^(-2/3)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                   
  /                   
 |                    
 |       __________   
 |      /      1      
 |     /  1 - ----  dx
 |    /        2/3    
 |  \/        x       
 |                    
/                     
0
∫011−1x23 dx\int\limits_{0}^{1} \sqrt{1 - \frac{1}{x^{\frac{2}{3}}}}\, dx 
Integral(sqrt(1 - 1/x^(2/3)), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                    
 |                          //      ___________           ___________                 \
 |      __________          ||     /       2/3     2/3   /       2/3        | 2/3|    |
 |     /      1             || - \/  -1 + x     + x   *\/  -1 + x       for |x   | > 1|
 |    /  1 - ----  dx = C + |<                                                        |
 |   /        2/3           ||       __________             __________                |
 | \/        x              ||      /      2/3       2/3   /      2/3                 |
 |                          \\- I*\/  1 - x     + I*x   *\/  1 - x        otherwise   /
/
∫1−1x23 dx=C+{x23x23−1−x23−1for ∣x23∣>1ix231−x23−i1−x23otherwise\int \sqrt{1 - \frac{1}{x^{\frac{2}{3}}}}\, dx = C + \begin{cases} x^{\frac{2}{3}} \sqrt{x^{\frac{2}{3}} - 1} - \sqrt{x^{\frac{2}{3}} - 1} & \text{for}\: \left|{x^{\frac{2}{3}}}\right| > 1 \\i x^{\frac{2}{3}} \sqrt{1 - x^{\frac{2}{3}}} - i \sqrt{1 - x^{\frac{2}{3}}} & \text{otherwise} \end{cases}
Simplificar
Gráfica
1.0000000.9997500.9997750.9998000.9998250.9998500.9998750.9999000.9999250.9999500.99997501
Trazar el gráfico f(x)
Respuesta [src] 
  1                                                                                 
  /                                                                                 
 |                                                                                  
 |  /                                                   ___________                 
 |  |                                3 ___             /       2/3                  
 |  |            1                   \/ x          2*\/  -1 + x           2/3       
 |  |- ---------------------- + ---------------- + ----------------  for x    > 1   
 |  |             ___________        ___________         3 ___                      
 |  |    3 ___   /       2/3        /       2/3        3*\/ x                       
 |  |  3*\/ x *\/  -1 + x       3*\/  -1 + x                                        
 |  <                                                                             dx
 |  |                                                   __________                  
 |  |        3 ___                                     /      2/3                   
 |  |      I*\/ x                  I             2*I*\/  1 - x                      
 |  |- --------------- + --------------------- + -----------------    otherwise     
 |  |       __________              __________          3 ___                       
 |  |      /      2/3      3 ___   /      2/3         3*\/ x                        
 |  \  3*\/  1 - x       3*\/ x *\/  1 - x                                          
 |                                                                                  
/                                                                                   
0
∫01{x33x23−1+2x23−13x3−13x3x23−1for x23>1−ix331−x23+2i1−x233x3+i3x31−x23otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{\sqrt[3]{x}}{3 \sqrt{x^{\frac{2}{3}} - 1}} + \frac{2 \sqrt{x^{\frac{2}{3}} - 1}}{3 \sqrt[3]{x}} - \frac{1}{3 \sqrt[3]{x} \sqrt{x^{\frac{2}{3}} - 1}} & \text{for}\: x^{\frac{2}{3}} > 1 \\- \frac{i \sqrt[3]{x}}{3 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{2 i \sqrt{1 - x^{\frac{2}{3}}}}{3 \sqrt[3]{x}} + \frac{i}{3 \sqrt[3]{x} \sqrt{1 - x^{\frac{2}{3}}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                 
  /                                                                                 
 |                                                                                  
 |  /                                                   ___________                 
 |  |                                3 ___             /       2/3                  
 |  |            1                   \/ x          2*\/  -1 + x           2/3       
 |  |- ---------------------- + ---------------- + ----------------  for x    > 1   
 |  |             ___________        ___________         3 ___                      
 |  |    3 ___   /       2/3        /       2/3        3*\/ x                       
 |  |  3*\/ x *\/  -1 + x       3*\/  -1 + x                                        
 |  <                                                                             dx
 |  |                                                   __________                  
 |  |        3 ___                                     /      2/3                   
 |  |      I*\/ x                  I             2*I*\/  1 - x                      
 |  |- --------------- + --------------------- + -----------------    otherwise     
 |  |       __________              __________          3 ___                       
 |  |      /      2/3      3 ___   /      2/3         3*\/ x                        
 |  \  3*\/  1 - x       3*\/ x *\/  1 - x                                          
 |                                                                                  
/                                                                                   
0
∫01{x33x23−1+2x23−13x3−13x3x23−1for x23>1−ix331−x23+2i1−x233x3+i3x31−x23otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{\sqrt[3]{x}}{3 \sqrt{x^{\frac{2}{3}} - 1}} + \frac{2 \sqrt{x^{\frac{2}{3}} - 1}}{3 \sqrt[3]{x}} - \frac{1}{3 \sqrt[3]{x} \sqrt{x^{\frac{2}{3}} - 1}} & \text{for}\: x^{\frac{2}{3}} > 1 \\- \frac{i \sqrt[3]{x}}{3 \sqrt{1 - x^{\frac{2}{3}}}} + \frac{2 i \sqrt{1 - x^{\frac{2}{3}}}}{3 \sqrt[3]{x}} + \frac{i}{3 \sqrt[3]{x} \sqrt{1 - x^{\frac{2}{3}}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-1/(3*x^(1/3)*sqrt(-1 + x^(2/3))) + x^(1/3)/(3*sqrt(-1 + x^(2/3))) + 2*sqrt(-1 + x^(2/3))/(3*x^(1/3)), x^(2/3) > 1), (-i*x^(1/3)/(3*sqrt(1 - x^(2/3))) + i/(3*x^(1/3)*sqrt(1 - x^(2/3))) + 2*i*sqrt(1 - x^(2/3))/(3*x^(1/3)), True)), (x, 0, 1))
Respuesta numérica [src] 
(0.0 + 0.999999999999689j)
(0.0 + 0.999999999999689j)

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

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