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Resolución de integrales/ x^2√(x^6-1)

Integral de x^2√(x^6-1) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                  
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 |   2   /  6        
 |  x *\/  x  - 1  dx
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0
∫01x2x6−1 dx\int\limits_{0}^{1} x^{2} \sqrt{x^{6} - 1}\, dx 
Integral(x^2*sqrt(x^6 - 1), (x, 0, 1))
Respuesta (Indefinida) [src] 
                           //       / 3\          3                9                    \
                           ||  acosh\x /         x                x             | 6|    |
  /                        ||- --------- - -------------- + --------------  for |x | > 1|
 |                         ||      6            _________        _________              |
 |       ________          ||                  /       6        /       6               |
 |  2   /  6               ||              6*\/  -1 + x     6*\/  -1 + x                |
 | x *\/  x  - 1  dx = C + |<                                                           |
 |                         ||                             ________                      |
/                          ||              / 3\      3   /      6                       |
                           ||        I*asin\x /   I*x *\/  1 - x                        |
                           ||        ---------- + ----------------           otherwise  |
                           ||            6               6                              |
                           \\                                                           /
∫x2x6−1 dx=C+{x96x6−1−x36x6−1−acosh⁡(x3)6for ∣x6∣>1ix31−x66+iasin⁡(x3)6otherwise\int x^{2} \sqrt{x^{6} - 1}\, dx = C + \begin{cases} \frac{x^{9}}{6 \sqrt{x^{6} - 1}} - \frac{x^{3}}{6 \sqrt{x^{6} - 1}} - \frac{\operatorname{acosh}{\left(x^{3} \right)}}{6} & \text{for}\: \left|{x^{6}}\right| > 1 \\\frac{i x^{3} \sqrt{1 - x^{6}}}{6} + \frac{i \operatorname{asin}{\left(x^{3} \right)}}{6} & \text{otherwise} \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.9001
Trazar el gráfico f(x)
Respuesta [src] 
  1                                                                                
  /                                                                                
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 |  /       8               2              14                 8                    
 |  |      x               x              x                3*x             6       
 |  |-------------- - ------------ - -------------- + --------------  for x  > 1   
 |  |           3/2      _________              3/2        _________               
 |  |  /      6\        /       6      /      6\          /       6                
 |  |2*\-1 + x /      \/  -1 + x     2*\-1 + x /      2*\/  -1 + x                 
 |  |                                                                              
 |  <               ________                                                     dx
 |  |          2   /      6            2               8                           
 |  |       I*x *\/  1 - x          I*x             I*x                            
 |  |       ---------------- + ------------- - -------------          otherwise    
 |  |              2                ________        ________                       
 |  |                              /      6        /      6                        
 |  |                          2*\/  1 - x     2*\/  1 - x                         
 |  \                                                                              
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/                                                                                  
0
∫01{−x142(x6−1)32+3x82x6−1+x82(x6−1)32−x2x6−1for x6>1−ix821−x6+ix21−x62+ix221−x6otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{x^{14}}{2 \left(x^{6} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{8}}{2 \sqrt{x^{6} - 1}} + \frac{x^{8}}{2 \left(x^{6} - 1\right)^{\frac{3}{2}}} - \frac{x^{2}}{\sqrt{x^{6} - 1}} & \text{for}\: x^{6} > 1 \\- \frac{i x^{8}}{2 \sqrt{1 - x^{6}}} + \frac{i x^{2} \sqrt{1 - x^{6}}}{2} + \frac{i x^{2}}{2 \sqrt{1 - x^{6}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                
  /                                                                                
 |                                                                                 
 |  /       8               2              14                 8                    
 |  |      x               x              x                3*x             6       
 |  |-------------- - ------------ - -------------- + --------------  for x  > 1   
 |  |           3/2      _________              3/2        _________               
 |  |  /      6\        /       6      /      6\          /       6                
 |  |2*\-1 + x /      \/  -1 + x     2*\-1 + x /      2*\/  -1 + x                 
 |  |                                                                              
 |  <               ________                                                     dx
 |  |          2   /      6            2               8                           
 |  |       I*x *\/  1 - x          I*x             I*x                            
 |  |       ---------------- + ------------- - -------------          otherwise    
 |  |              2                ________        ________                       
 |  |                              /      6        /      6                        
 |  |                          2*\/  1 - x     2*\/  1 - x                         
 |  \                                                                              
 |                                                                                 
/                                                                                  
0
∫01{−x142(x6−1)32+3x82x6−1+x82(x6−1)32−x2x6−1for x6>1−ix821−x6+ix21−x62+ix221−x6otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{x^{14}}{2 \left(x^{6} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{8}}{2 \sqrt{x^{6} - 1}} + \frac{x^{8}}{2 \left(x^{6} - 1\right)^{\frac{3}{2}}} - \frac{x^{2}}{\sqrt{x^{6} - 1}} & \text{for}\: x^{6} > 1 \\- \frac{i x^{8}}{2 \sqrt{1 - x^{6}}} + \frac{i x^{2} \sqrt{1 - x^{6}}}{2} + \frac{i x^{2}}{2 \sqrt{1 - x^{6}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((x^8/(2*(-1 + x^6)^(3/2)) - x^2/sqrt(-1 + x^6) - x^14/(2*(-1 + x^6)^(3/2)) + 3*x^8/(2*sqrt(-1 + x^6)), x^6 > 1), (i*x^2*sqrt(1 - x^6)/2 + i*x^2/(2*sqrt(1 - x^6)) - i*x^8/(2*sqrt(1 - x^6)), True)), (x, 0, 1))
Respuesta numérica [src] 
(0.0 + 0.261799387799149j)
(0.0 + 0.261799387799149j)

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

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