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

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

Solución

Ha introducido [src]

  c                      
  /                      
 |                       
 |        ____________   
 |       /       2  2    
 |      /   2   x *a     
 |     /   a  - -----  dx
 |    /            2     
 |  \/            b      
 |                       
/                        
0

\int\limits_{0}^{c} \sqrt{a^{2} - \frac{a^{2} x^{2}}{b^{2}}}\, dx

Integral(sqrt(a^2 - x^2*a^2/b^2), (x, 0, c))

Respuesta (Indefinida) [src]

                               //             /x\                                                         \
                               ||  I*a*b*acosh|-|                                   3             | 2|    |
                               ||             \b/         I*a*x                I*a*x              |x |    |
                               ||- -------------- - ----------------- + --------------------  for |--| > 1|
  /                            ||        2                  _________              _________      | 2|    |
 |                             ||                          /       2              /       2       |b |    |
 |       ____________          ||                         /       x        2     /       x                |
 |      /       2  2           ||                   2*   /   -1 + --    2*b *   /   -1 + --               |
 |     /   2   x *a            ||                       /          2           /          2               |
 |    /   a  - -----  dx = C + |<                     \/          b          \/          b                |
 |   /            2            ||                                                                         |
 | \/            b             ||                                     ________                            |
 |                             ||                                    /      2                             |
/                              ||                                   /      x                              |
                               ||                     /x\   a*x*   /   1 - --                             |
                               ||             a*b*asin|-|         /         2                             |
                               ||                     \b/       \/         b                              |
                               ||             ----------- + ------------------                 otherwise  |
                               \\                  2                2                                     /

\int \sqrt{a^{2} - \frac{a^{2} x^{2}}{b^{2}}}\, dx = C + \begin{cases} - \frac{i a b \operatorname{acosh}{\left(\frac{x}{b} \right)}}{2} - \frac{i a x}{2 \sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{i a x^{3}}{2 b^{2} \sqrt{-1 + \frac{x^{2}}{b^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{b^{2}}}\right| > 1 \\\frac{a b \operatorname{asin}{\left(\frac{x}{b} \right)}}{2} + \frac{a x \sqrt{1 - \frac{x^{2}}{b^{2}}}}{2} & \text{otherwise} \end{cases}

Simplificar

Respuesta [src]

/   0                                                                                                              
|   /                                                                                                              
|  |                                                                                                               
|  |  /                               2                   4                     2            | 2|                  
|  |  |        I*a               I*a*x               I*a*x               3*I*a*x             |x |                  
|  |  |- --------------- + ----------------- - ----------------- + --------------------  for |--| > 1              
|  |  |        _________                 3/2                 3/2              _________      | 2|                  
|  |  |       /       2         /      2\           /      2\                /       2       |b |                  
|  |  |      /       x        2 |     x |         4 |     x |         2     /       x                              
|  |  |     /   -1 + --    2*b *|-1 + --|      2*b *|-1 + --|      2*b *   /   -1 + --                             
|  |  |    /          2         |      2|           |      2|             /          2                             
|  |  |  \/          b          \     b /           \     b /           \/          b                              
|  |  |                                                                                                            
|  |  |                   ________                                                                                 
|- |  <                  /      2                                                                     dx  for c < 0
|  |  |                 /      x                                                                                   
|  |  |           a*   /   1 - --                                                                                  
|  |  |               /         2                                  2                                               
|  |  |             \/         b            a                   a*x                                                
|  |  |           ---------------- + ---------------- - -------------------               otherwise                
|  |  |                  2                   ________              ________                                        
|  |  |                                     /      2              /      2                                         
|  |  |                                    /      x        2     /      x                                          
|  |  |                              2*   /   1 - --    2*b *   /   1 - --                                         
|  |  |                                  /         2           /         2                                         
|  |  \                                \/         b          \/         b                                          
|  |                                                                                                               
| /                                                                                                                
| c                                                                                                                
<                                                                                                                  
|  c                                                                                                               
|  /                                                                                                               
| |                                                                                                                
| |  /                               2                   4                     2            | 2|                   
| |  |        I*a               I*a*x               I*a*x               3*I*a*x             |x |                   
| |  |- --------------- + ----------------- - ----------------- + --------------------  for |--| > 1               
| |  |        _________                 3/2                 3/2              _________      | 2|                   
| |  |       /       2         /      2\           /      2\                /       2       |b |                   
| |  |      /       x        2 |     x |         4 |     x |         2     /       x                               
| |  |     /   -1 + --    2*b *|-1 + --|      2*b *|-1 + --|      2*b *   /   -1 + --                              
| |  |    /          2         |      2|           |      2|             /          2                              
| |  |  \/          b          \     b /           \     b /           \/          b                               
| |  |                                                                                                             
| |  |                   ________                                                                                  
| |  <                  /      2                                                                     dx   otherwise
| |  |                 /      x                                                                                    
| |  |           a*   /   1 - --                                                                                   
| |  |               /         2                                  2                                                
| |  |             \/         b            a                   a*x                                                 
| |  |           ---------------- + ---------------- - -------------------               otherwise                 
| |  |                  2                   ________              ________                                         
| |  |                                     /      2              /      2                                          
| |  |                                    /      x        2     /      x                                           
| |  |                              2*   /   1 - --    2*b *   /   1 - --                                          
| |  |                                  /         2           /         2                                          
| |  \                                \/         b          \/         b                                           
| |                                                                                                                
|/                                                                                                                 
\0

\begin{cases} - \int\limits_{c}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{3 i a x^{2}}{2 b^{2} \sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{i a x^{2}}{2 b^{2} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} - \frac{i a x^{4}}{2 b^{4} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{b^{2}}}\right| > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{b^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{b^{2}}}} - \frac{a x^{2}}{2 b^{2} \sqrt{1 - \frac{x^{2}}{b^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: c < 0 \\\int\limits_{0}^{c} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{3 i a x^{2}}{2 b^{2} \sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{i a x^{2}}{2 b^{2} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} - \frac{i a x^{4}}{2 b^{4} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{b^{2}}}\right| > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{b^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{b^{2}}}} - \frac{a x^{2}}{2 b^{2} \sqrt{1 - \frac{x^{2}}{b^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}

/   0                                                                                                              
|   /                                                                                                              
|  |                                                                                                               
|  |  /                               2                   4                     2            | 2|                  
|  |  |        I*a               I*a*x               I*a*x               3*I*a*x             |x |                  
|  |  |- --------------- + ----------------- - ----------------- + --------------------  for |--| > 1              
|  |  |        _________                 3/2                 3/2              _________      | 2|                  
|  |  |       /       2         /      2\           /      2\                /       2       |b |                  
|  |  |      /       x        2 |     x |         4 |     x |         2     /       x                              
|  |  |     /   -1 + --    2*b *|-1 + --|      2*b *|-1 + --|      2*b *   /   -1 + --                             
|  |  |    /          2         |      2|           |      2|             /          2                             
|  |  |  \/          b          \     b /           \     b /           \/          b                              
|  |  |                                                                                                            
|  |  |                   ________                                                                                 
|- |  <                  /      2                                                                     dx  for c < 0
|  |  |                 /      x                                                                                   
|  |  |           a*   /   1 - --                                                                                  
|  |  |               /         2                                  2                                               
|  |  |             \/         b            a                   a*x                                                
|  |  |           ---------------- + ---------------- - -------------------               otherwise                
|  |  |                  2                   ________              ________                                        
|  |  |                                     /      2              /      2                                         
|  |  |                                    /      x        2     /      x                                          
|  |  |                              2*   /   1 - --    2*b *   /   1 - --                                         
|  |  |                                  /         2           /         2                                         
|  |  \                                \/         b          \/         b                                          
|  |                                                                                                               
| /                                                                                                                
| c                                                                                                                
<                                                                                                                  
|  c                                                                                                               
|  /                                                                                                               
| |                                                                                                                
| |  /                               2                   4                     2            | 2|                   
| |  |        I*a               I*a*x               I*a*x               3*I*a*x             |x |                   
| |  |- --------------- + ----------------- - ----------------- + --------------------  for |--| > 1               
| |  |        _________                 3/2                 3/2              _________      | 2|                   
| |  |       /       2         /      2\           /      2\                /       2       |b |                   
| |  |      /       x        2 |     x |         4 |     x |         2     /       x                               
| |  |     /   -1 + --    2*b *|-1 + --|      2*b *|-1 + --|      2*b *   /   -1 + --                              
| |  |    /          2         |      2|           |      2|             /          2                              
| |  |  \/          b          \     b /           \     b /           \/          b                               
| |  |                                                                                                             
| |  |                   ________                                                                                  
| |  <                  /      2                                                                     dx   otherwise
| |  |                 /      x                                                                                    
| |  |           a*   /   1 - --                                                                                   
| |  |               /         2                                  2                                                
| |  |             \/         b            a                   a*x                                                 
| |  |           ---------------- + ---------------- - -------------------               otherwise                 
| |  |                  2                   ________              ________                                         
| |  |                                     /      2              /      2                                          
| |  |                                    /      x        2     /      x                                           
| |  |                              2*   /   1 - --    2*b *   /   1 - --                                          
| |  |                                  /         2           /         2                                          
| |  \                                \/         b          \/         b                                           
| |                                                                                                                
|/                                                                                                                 
\0

\begin{cases} - \int\limits_{c}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{3 i a x^{2}}{2 b^{2} \sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{i a x^{2}}{2 b^{2} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} - \frac{i a x^{4}}{2 b^{4} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{b^{2}}}\right| > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{b^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{b^{2}}}} - \frac{a x^{2}}{2 b^{2} \sqrt{1 - \frac{x^{2}}{b^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: c < 0 \\\int\limits_{0}^{c} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{3 i a x^{2}}{2 b^{2} \sqrt{-1 + \frac{x^{2}}{b^{2}}}} + \frac{i a x^{2}}{2 b^{2} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} - \frac{i a x^{4}}{2 b^{4} \left(-1 + \frac{x^{2}}{b^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{x^{2}}{b^{2}}}\right| > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{b^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{b^{2}}}} - \frac{a x^{2}}{2 b^{2} \sqrt{1 - \frac{x^{2}}{b^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}

Piecewise((-Integral(Piecewise((-i*a/sqrt(-1 + x^2/b^2) + i*a*x^2/(2*b^2*(-1 + x^2/b^2)^(3/2)) - i*a*x^4/(2*b^4*(-1 + x^2/b^2)^(3/2)) + 3*i*a*x^2/(2*b^2*sqrt(-1 + x^2/b^2)), |x^2/b^2| > 1), (a*sqrt(1 - x^2/b^2)/2 + a/(2*sqrt(1 - x^2/b^2)) - a*x^2/(2*b^2*sqrt(1 - x^2/b^2)), True)), (x, c, 0)), c < 0), (Integral(Piecewise((-i*a/sqrt(-1 + x^2/b^2) + i*a*x^2/(2*b^2*(-1 + x^2/b^2)^(3/2)) - i*a*x^4/(2*b^4*(-1 + x^2/b^2)^(3/2)) + 3*i*a*x^2/(2*b^2*sqrt(-1 + x^2/b^2)), |x^2/b^2| > 1), (a*sqrt(1 - x^2/b^2)/2 + a/(2*sqrt(1 - x^2/b^2)) - a*x^2/(2*b^2*sqrt(1 - x^2/b^2)), True)), (x, 0, c)), True))

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

Otras calculadoras

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Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

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

Límites de integración:

Gráfico:

Definida a trozos:

Solución