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Integral de dx/sqrt(a^2-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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