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Integral de 5x/(sqrt(16-x^4)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  2                
  /                
 |                 
 |      5*x        
 |  ------------ dx
 |     _________   
 |    /       4    
 |  \/  16 - x     
 |                 
/                  
0                  
$$\int\limits_{0}^{2} \frac{5 x}{\sqrt{16 - x^{4}}}\, dx$$
Integral((5*x)/sqrt(16 - x^4), (x, 0, 2))
Respuesta (Indefinida) [src]
                         //          / 2\              \
                         ||          |x |              |
                         ||-5*I*acosh|--|      | 4|    |
  /                      ||          \4 /      |x |    |
 |                       ||--------------  for ---- > 1|
 |     5*x               ||      2              16     |
 | ------------ dx = C + |<                            |
 |    _________          ||        / 2\                |
 |   /       4           ||        |x |                |
 | \/  16 - x            ||  5*asin|--|                |
 |                       ||        \4 /                |
/                        ||  ----------     otherwise  |
                         \\      2                     /
$$\int \frac{5 x}{\sqrt{16 - x^{4}}}\, dx = C + \begin{cases} - \frac{5 i \operatorname{acosh}{\left(\frac{x^{2}}{4} \right)}}{2} & \text{for}\: \frac{\left|{x^{4}}\right|}{16} > 1 \\\frac{5 \operatorname{asin}{\left(\frac{x^{2}}{4} \right)}}{2} & \text{otherwise} \end{cases}$$
Gráfica
Respuesta [src]
  2                                 
  /                                 
 |                                  
 |  /                       4       
 |  |     -5*I*x           x        
 |  |----------------  for -- > 1   
 |  |       _________      16       
 |  |      /       4                
 |  |     /       x                 
 |  |4*  /   -1 + --                
 |  |  \/         16                
 |  <                             dx
 |  |      5*x                      
 |  |---------------   otherwise    
 |  |       ________                
 |  |      /      4                 
 |  |     /      x                  
 |  |4*  /   1 - --                 
 |  |  \/        16                 
 |  \                               
 |                                  
/                                   
0                                   
$$\int\limits_{0}^{2} \begin{cases} - \frac{5 i x}{4 \sqrt{\frac{x^{4}}{16} - 1}} & \text{for}\: \frac{x^{4}}{16} > 1 \\\frac{5 x}{4 \sqrt{1 - \frac{x^{4}}{16}}} & \text{otherwise} \end{cases}\, dx$$
=
=
  2                                 
  /                                 
 |                                  
 |  /                       4       
 |  |     -5*I*x           x        
 |  |----------------  for -- > 1   
 |  |       _________      16       
 |  |      /       4                
 |  |     /       x                 
 |  |4*  /   -1 + --                
 |  |  \/         16                
 |  <                             dx
 |  |      5*x                      
 |  |---------------   otherwise    
 |  |       ________                
 |  |      /      4                 
 |  |     /      x                  
 |  |4*  /   1 - --                 
 |  |  \/        16                 
 |  \                               
 |                                  
/                                   
0                                   
$$\int\limits_{0}^{2} \begin{cases} - \frac{5 i x}{4 \sqrt{\frac{x^{4}}{16} - 1}} & \text{for}\: \frac{x^{4}}{16} > 1 \\\frac{5 x}{4 \sqrt{1 - \frac{x^{4}}{16}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-5*i*x/(4*sqrt(-1 + x^4/16)), x^4/16 > 1), (5*x/(4*sqrt(1 - x^4/16)), True)), (x, 0, 2))
Respuesta numérica [src]
3.92699081531275
3.92699081531275

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