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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1               
  /               
 |                
 |       x        
 |  ----------- dx
 |     ________   
 |    /      4    
 |  \/  2 - x     
 |                
/                 
0                 
$$\int\limits_{0}^{1} \frac{x}{\sqrt{2 - x^{4}}}\, dx$$
Integral(x/sqrt(2 - x^4), (x, 0, 1))
Respuesta (Indefinida) [src]
                        //        /  ___  2\               \
                        ||        |\/ 2 *x |               |
                        ||-I*acosh|--------|       | 4|    |
  /                     ||        \   2    /       |x |    |
 |                      ||-------------------  for ---- > 1|
 |      x               ||         2                2      |
 | ----------- dx = C + |<                                 |
 |    ________          ||      /  ___  2\                 |
 |   /      4           ||      |\/ 2 *x |                 |
 | \/  2 - x            ||  asin|--------|                 |
 |                      ||      \   2    /                 |
/                       ||  --------------      otherwise  |
                        \\        2                        /
$$\int \frac{x}{\sqrt{2 - x^{4}}}\, dx = C + \begin{cases} - \frac{i \operatorname{acosh}{\left(\frac{\sqrt{2} x^{2}}{2} \right)}}{2} & \text{for}\: \frac{\left|{x^{4}}\right|}{2} > 1 \\\frac{\operatorname{asin}{\left(\frac{\sqrt{2} x^{2}}{2} \right)}}{2} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta [src]
  1                                 
  /                                 
 |                                  
 |  /         ___           4       
 |  |  -I*x*\/ 2           x        
 |  |----------------  for -- > 1   
 |  |       _________      2        
 |  |      /       4                
 |  |     /       x                 
 |  |2*  /   -1 + --                
 |  |  \/         2                 
 |  <                             dx
 |  |        ___                    
 |  |    x*\/ 2                     
 |  |---------------   otherwise    
 |  |       ________                
 |  |      /      4                 
 |  |     /      x                  
 |  |2*  /   1 - --                 
 |  \  \/        2                  
 |                                  
/                                   
0                                   
$$\int\limits_{0}^{1} \begin{cases} - \frac{\sqrt{2} i x}{2 \sqrt{\frac{x^{4}}{2} - 1}} & \text{for}\: \frac{x^{4}}{2} > 1 \\\frac{\sqrt{2} x}{2 \sqrt{1 - \frac{x^{4}}{2}}} & \text{otherwise} \end{cases}\, dx$$
=
=
  1                                 
  /                                 
 |                                  
 |  /         ___           4       
 |  |  -I*x*\/ 2           x        
 |  |----------------  for -- > 1   
 |  |       _________      2        
 |  |      /       4                
 |  |     /       x                 
 |  |2*  /   -1 + --                
 |  |  \/         2                 
 |  <                             dx
 |  |        ___                    
 |  |    x*\/ 2                     
 |  |---------------   otherwise    
 |  |       ________                
 |  |      /      4                 
 |  |     /      x                  
 |  |2*  /   1 - --                 
 |  \  \/        2                  
 |                                  
/                                   
0                                   
$$\int\limits_{0}^{1} \begin{cases} - \frac{\sqrt{2} i x}{2 \sqrt{\frac{x^{4}}{2} - 1}} & \text{for}\: \frac{x^{4}}{2} > 1 \\\frac{\sqrt{2} x}{2 \sqrt{1 - \frac{x^{4}}{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*x*sqrt(2)/(2*sqrt(-1 + x^4/2)), x^4/2 > 1), (x*sqrt(2)/(2*sqrt(1 - x^4/2)), True)), (x, 0, 1))
Respuesta numérica [src]
0.392699081698724
0.392699081698724

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