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

Integral de sqrt(x^2-y^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  0                
  /                
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 |     _________   
 |    /  2    2    
 |  \/  x  - y   dx
 |                 
/                  
-r
∫−r0x2−y2 dx\int\limits_{- r}^{0} \sqrt{x^{2} - y^{2}}\, dx 
Integral(sqrt(x^2 - y^2), (x, -r, 0))
Respuesta (Indefinida) [src] 
                         //   2      /x\                                                        \
                         ||  y *acosh|-|             3                                  | 2|    |
                         ||          \y/            x                   x*y             |x |    |
                         ||- ----------- + ------------------- - -----------------  for |--| > 1|
                         ||       2                  _________           _________      | 2|    |
  /                      ||                         /       2           /       2       |y |    |
 |                       ||                        /       x           /       x                |
 |    _________          ||                2*y*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                      /          2        /          2               |
 | \/  x  - y   dx = C + |<                    \/          y       \/          y                |
 |                       ||                                                                     |
/                        ||                                     ________                        |
                         ||                                    /      2                         |
                         ||                                   /      x                          |
                         ||             2     /x\   I*x*y*   /   1 - --                         |
                         ||          I*y *asin|-|           /         2                         |
                         ||                   \y/         \/         y                          |
                         ||          ------------ + --------------------             otherwise  |
                         \\               2                  2                                  /
∫x2−y2 dx=C+{x32yx2y2−1−xy2x2y2−1−y2acosh⁡(xy)2for ∣x2y2∣>1ixy−x2y2+12+iy2asin⁡(xy)2otherwise\int \sqrt{x^{2} - y^{2}}\, dx = C + \begin{cases} \frac{x^{3}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{x y}{2 \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{y^{2} \operatorname{acosh}{\left(\frac{x}{y} \right)}}{2} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\\frac{i x y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y^{2} \operatorname{asin}{\left(\frac{x}{y} \right)}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/  0                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                            2                   4                     2             | 2|                   
| |  |         y                 x                   x                   3*x              |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for |--| > 1               
| |  |        _________                3/2                 3/2             _________      | 2|                   
| |  |       /       2        /      2\           /      2\               /       2       |y |                   
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*y*|-1 + --|      2*y *|-1 + --|      2*y*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          y         \     y /           \     y /          \/          y                               
| |  |                                                                                                           
| |  |                    ________                                                                               
| |  <                   /      2                                                                  dx   for r > 0
| |  |                  /      x                                                                                 
| |  |          I*y*   /   1 - --                                                                                
| |  |                /         2                                 2                                              
| |  |              \/         y           I*y                 I*x                                               
| |  |          ------------------ + ---------------- - ------------------             otherwise                 
| |  |                  2                    ________             ________                                       
| |  |                                      /      2             /      2                                        
| |  |                                     /      x             /      x                                         
| |  |                               2*   /   1 - --    2*y*   /   1 - --                                        
| |  |                                   /         2          /         2                                        
| |  \                                 \/         y         \/         y                                         
| |                                                                                                              
|/                                                                                                               
|-r                                                                                                              
<                                                                                                                
|  -r                                                                                                            
|   /                                                                                                            
|  |                                                                                                             
|  |  /                            2                   4                     2             | 2|                  
|  |  |         y                 x                   x                   3*x              |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for |--| > 1              
|  |  |        _________                3/2                 3/2             _________      | 2|                  
|  |  |       /       2        /      2\           /      2\               /       2       |y |                  
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*y*|-1 + --|      2*y *|-1 + --|      2*y*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          y         \     y /           \     y /          \/          y                              
|  |  |                                                                                                          
|  |  |                    ________                                                                              
|- |  <                   /      2                                                                  dx  otherwise
|  |  |                  /      x                                                                                
|  |  |          I*y*   /   1 - --                                                                               
|  |  |                /         2                                 2                                             
|  |  |              \/         y           I*y                 I*x                                              
|  |  |          ------------------ + ---------------- - ------------------             otherwise                
|  |  |                  2                    ________             ________                                      
|  |  |                                      /      2             /      2                                       
|  |  |                                     /      x             /      x                                        
|  |  |                               2*   /   1 - --    2*y*   /   1 - --                                       
|  |  |                                   /         2          /         2                                       
|  |  \                                 \/         y         \/         y                                        
|  |                                                                                                             
| /                                                                                                              
\ 0
{∫−r0{−x42y3(x2y2−1)32+3x22yx2y2−1+x22y(x2y2−1)32−yx2y2−1for ∣x2y2∣>1−ix22y−x2y2+1+iy−x2y2+12+iy2−x2y2+1otherwise dxfor r>0−∫0−r{−x42y3(x2y2−1)32+3x22yx2y2−1+x22y(x2y2−1)32−yx2y2−1for ∣x2y2∣>1−ix22y−x2y2+1+iy−x2y2+12+iy2−x2y2+1otherwise dxotherwise\begin{cases} \int\limits_{- r}^{0} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r > 0 \\- \int\limits_{0}^{- r} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
=
= 
/  0                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                            2                   4                     2             | 2|                   
| |  |         y                 x                   x                   3*x              |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for |--| > 1               
| |  |        _________                3/2                 3/2             _________      | 2|                   
| |  |       /       2        /      2\           /      2\               /       2       |y |                   
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*y*|-1 + --|      2*y *|-1 + --|      2*y*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          y         \     y /           \     y /          \/          y                               
| |  |                                                                                                           
| |  |                    ________                                                                               
| |  <                   /      2                                                                  dx   for r > 0
| |  |                  /      x                                                                                 
| |  |          I*y*   /   1 - --                                                                                
| |  |                /         2                                 2                                              
| |  |              \/         y           I*y                 I*x                                               
| |  |          ------------------ + ---------------- - ------------------             otherwise                 
| |  |                  2                    ________             ________                                       
| |  |                                      /      2             /      2                                        
| |  |                                     /      x             /      x                                         
| |  |                               2*   /   1 - --    2*y*   /   1 - --                                        
| |  |                                   /         2          /         2                                        
| |  \                                 \/         y         \/         y                                         
| |                                                                                                              
|/                                                                                                               
|-r                                                                                                              
<                                                                                                                
|  -r                                                                                                            
|   /                                                                                                            
|  |                                                                                                             
|  |  /                            2                   4                     2             | 2|                  
|  |  |         y                 x                   x                   3*x              |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for |--| > 1              
|  |  |        _________                3/2                 3/2             _________      | 2|                  
|  |  |       /       2        /      2\           /      2\               /       2       |y |                  
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*y*|-1 + --|      2*y *|-1 + --|      2*y*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          y         \     y /           \     y /          \/          y                              
|  |  |                                                                                                          
|  |  |                    ________                                                                              
|- |  <                   /      2                                                                  dx  otherwise
|  |  |                  /      x                                                                                
|  |  |          I*y*   /   1 - --                                                                               
|  |  |                /         2                                 2                                             
|  |  |              \/         y           I*y                 I*x                                              
|  |  |          ------------------ + ---------------- - ------------------             otherwise                
|  |  |                  2                    ________             ________                                      
|  |  |                                      /      2             /      2                                       
|  |  |                                     /      x             /      x                                        
|  |  |                               2*   /   1 - --    2*y*   /   1 - --                                       
|  |  |                                   /         2          /         2                                       
|  |  \                                 \/         y         \/         y                                        
|  |                                                                                                             
| /                                                                                                              
\ 0
{∫−r0{−x42y3(x2y2−1)32+3x22yx2y2−1+x22y(x2y2−1)32−yx2y2−1for ∣x2y2∣>1−ix22y−x2y2+1+iy−x2y2+12+iy2−x2y2+1otherwise dxfor r>0−∫0−r{−x42y3(x2y2−1)32+3x22yx2y2−1+x22y(x2y2−1)32−yx2y2−1for ∣x2y2∣>1−ix22y−x2y2+1+iy−x2y2+12+iy2−x2y2+1otherwise dxotherwise\begin{cases} \int\limits_{- r}^{0} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r > 0 \\- \int\limits_{0}^{- r} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
Piecewise((Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, -r, 0)), r > 0), (-Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, 0, -r)), True))

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

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