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Resolución de integrales/ sqrt(x^2-a^2)/ sqrt(x^2-a^2)/x

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                
  /                
 |                 
 |     _________   
 |    /  2    2    
 |  \/  x  - a     
 |  ------------ dx
 |       x         
 |                 
/                  
0
∫01−a2+x2x dx\int\limits_{0}^{1} \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx 
Integral(sqrt(x^2 - a^2)/x, (x, 0, 1))
Respuesta (Indefinida) [src] 
                         //                                             2            | 2|    \
                         ||           /a\         I*x                I*a             |a |    |
                         ||- I*a*acosh|-| - --------------- + -----------------  for |--| > 1|
                         ||           \x/         _________           _________      | 2|    |
  /                      ||                      /       2           /       2       |x |    |
 |                       ||                     /       a           /       a                |
 |    _________          ||                    /   -1 + --    x*   /   -1 + --               |
 |   /  2    2           ||                   /          2        /          2               |
 | \/  x  - a            ||                 \/          x       \/          x                |
 | ------------ dx = C + |<                                                                  |
 |      x                ||                                        2                         |
 |                       ||         /a\         x                 a                          |
/                        ||   a*asin|-| + -------------- - ----------------       otherwise  |
                         ||         \x/         ________           ________                  |
                         ||                    /      2           /      2                   |
                         ||                   /      a           /      a                    |
                         ||                  /   1 - --    x*   /   1 - --                   |
                         ||                 /         2        /         2                   |
                         \\               \/         x       \/         x                    /
∫−a2+x2x dx=C+{ia2xa2x2−1−iaacosh⁡(ax)−ixa2x2−1for ∣a2x2∣>1−a2x−a2x2+1+aasin⁡(ax)+x−a2x2+1otherwise\int \frac{\sqrt{- a^{2} + x^{2}}}{x}\, dx = C + \begin{cases} \frac{i a^{2}}{x \sqrt{\frac{a^{2}}{x^{2}} - 1}} - i a \operatorname{acosh}{\left(\frac{a}{x} \right)} - \frac{i x}{\sqrt{\frac{a^{2}}{x^{2}} - 1}} & \text{for}\: \left|{\frac{a^{2}}{x^{2}}}\right| > 1 \\- \frac{a^{2}}{x \sqrt{- \frac{a^{2}}{x^{2}} + 1}} + a \operatorname{asin}{\left(\frac{a}{x} \right)} + \frac{x}{\sqrt{- \frac{a^{2}}{x^{2}} + 1}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                    
  /                                                                                    
 |                                                                                     
 |  /         3                              2                2             | 2|       
 |  |      I*x           2*I*x          I*x*a              I*a              |a |       
 |  |- ------------ - ------------ + ------------ + ------------------  for ---- > 1   
 |  |           3/2      _________            3/2            _________        2        
 |  |  / 2    2\        /  2    2    / 2    2\              /       2        x         
 |  |  \a  - x /      \/  a  - x     \a  - x /       2     /       a                   
 |  |                                               x *   /   -1 + --                  
 |  |                                                    /          2                  
 |  |                                                  \/          x                   
 |  <                                                                                dx
 |  |        3                              2                2                         
 |  |       x             2*x            x*a                a                          
 |  |- ------------ + ------------ + ------------ - -----------------    otherwise     
 |  |           3/2      _________            3/2            ________                  
 |  |  / 2    2\        /  2    2    / 2    2\              /      2                   
 |  |  \x  - a /      \/  x  - a     \x  - a /       2     /      a                    
 |  |                                               x *   /   1 - --                   
 |  |                                                    /         2                   
 |  \                                                  \/         x                    
 |                                                                                     
/                                                                                      
0
∫01{ia2x(a2−x2)32+ia2x2a2x2−1−ix3(a2−x2)32−2ixa2−x2for ∣a2∣x2>1a2x(−a2+x2)32−a2x2−a2x2+1−x3(−a2+x2)32+2x−a2+x2otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{i a^{2} x}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} + \frac{i a^{2}}{x^{2} \sqrt{\frac{a^{2}}{x^{2}} - 1}} - \frac{i x^{3}}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} - \frac{2 i x}{\sqrt{a^{2} - x^{2}}} & \text{for}\: \frac{\left|{a^{2}}\right|}{x^{2}} > 1 \\\frac{a^{2} x}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \sqrt{- \frac{a^{2}}{x^{2}} + 1}} - \frac{x^{3}}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} + \frac{2 x}{\sqrt{- a^{2} + x^{2}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                    
  /                                                                                    
 |                                                                                     
 |  /         3                              2                2             | 2|       
 |  |      I*x           2*I*x          I*x*a              I*a              |a |       
 |  |- ------------ - ------------ + ------------ + ------------------  for ---- > 1   
 |  |           3/2      _________            3/2            _________        2        
 |  |  / 2    2\        /  2    2    / 2    2\              /       2        x         
 |  |  \a  - x /      \/  a  - x     \a  - x /       2     /       a                   
 |  |                                               x *   /   -1 + --                  
 |  |                                                    /          2                  
 |  |                                                  \/          x                   
 |  <                                                                                dx
 |  |        3                              2                2                         
 |  |       x             2*x            x*a                a                          
 |  |- ------------ + ------------ + ------------ - -----------------    otherwise     
 |  |           3/2      _________            3/2            ________                  
 |  |  / 2    2\        /  2    2    / 2    2\              /      2                   
 |  |  \x  - a /      \/  x  - a     \x  - a /       2     /      a                    
 |  |                                               x *   /   1 - --                   
 |  |                                                    /         2                   
 |  \                                                  \/         x                    
 |                                                                                     
/                                                                                      
0
∫01{ia2x(a2−x2)32+ia2x2a2x2−1−ix3(a2−x2)32−2ixa2−x2for ∣a2∣x2>1a2x(−a2+x2)32−a2x2−a2x2+1−x3(−a2+x2)32+2x−a2+x2otherwise dx\int\limits_{0}^{1} \begin{cases} \frac{i a^{2} x}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} + \frac{i a^{2}}{x^{2} \sqrt{\frac{a^{2}}{x^{2}} - 1}} - \frac{i x^{3}}{\left(a^{2} - x^{2}\right)^{\frac{3}{2}}} - \frac{2 i x}{\sqrt{a^{2} - x^{2}}} & \text{for}\: \frac{\left|{a^{2}}\right|}{x^{2}} > 1 \\\frac{a^{2} x}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} - \frac{a^{2}}{x^{2} \sqrt{- \frac{a^{2}}{x^{2}} + 1}} - \frac{x^{3}}{\left(- a^{2} + x^{2}\right)^{\frac{3}{2}}} + \frac{2 x}{\sqrt{- a^{2} + x^{2}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-i*x^3/(a^2 - x^2)^(3/2) - 2*i*x/sqrt(a^2 - x^2) + i*x*a^2/(a^2 - x^2)^(3/2) + i*a^2/(x^2*sqrt(-1 + a^2/x^2)), |a^2|/x^2 > 1), (-x^3/(x^2 - a^2)^(3/2) + 2*x/sqrt(x^2 - a^2) + x*a^2/(x^2 - a^2)^(3/2) - a^2/(x^2*sqrt(1 - a^2/x^2)), True)), (x, 0, 1))

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

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