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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  a                
  /                
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 |     _________   
 |    /  2    2    
 |  \/  a  - x   dx
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0
∫0aa2−x2 dx\int\limits_{0}^{a} \sqrt{a^{2} - x^{2}}\, dx 
Integral(sqrt(a^2 - x^2), (x, 0, a))
Respuesta (Indefinida) [src] 
                         //     2      /x\                                                        \
                         ||  I*a *acosh|-|              3                                 | 2|    |
                         ||            \a/           I*x                 I*a*x            |x |    |
                         ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||        2                   _________           _________      | 2|    |
  /                      ||                           /       2           /       2       |a |    |
 |                       ||                          /       x           /       x                |
 |    _________          ||                  2*a*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                        /          2        /          2               |
 | \/  a  - x   dx = C + |<                      \/          a       \/          a                |
 |                       ||                                                                       |
/                        ||                                    ________                           |
                         ||                                   /      2                            |
                         ||                                  /      x                             |
                         ||              2     /x\   a*x*   /   1 - --                            |
                         ||             a *asin|-|         /         2                            |
                         ||                    \a/       \/         a                             |
                         ||             ---------- + ------------------                otherwise  |
                         \\                 2                2                                    /
∫a2−x2 dx=C+{−ia2acosh⁡(xa)2−iax2−1+x2a2+ix32a−1+x2a2for ∣x2a2∣>1a2asin⁡(xa)2+ax1−x2a22otherwise\int \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{i a x}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{3}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} + \frac{a x \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/   0                                                                                                            
|   /                                                                                                            
|  |                                                                                                             
|  |  /                             2                   4                     2            | 2|                  
|  |  |        I*a               I*x                 I*x                 3*I*x             |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1              
|  |  |        _________                3/2                 3/2             _________        2                   
|  |  |       /       2        /      2\           /      2\               /       2        a                    
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          a         \     a /           \     a /          \/          a                              
|  |  |                                                                                                          
|  |  |                   ________                                                                               
|- |  <                  /      2                                                                   dx  for a < 0
|  |  |                 /      x                                                                                 
|  |  |           a*   /   1 - --                                                                                
|  |  |               /         2                                2                                               
|  |  |             \/         a            a                   x                                                
|  |  |           ---------------- + ---------------- - ------------------              otherwise                
|  |  |                  2                   ________             ________                                       
|  |  |                                     /      2             /      2                                        
|  |  |                                    /      x             /      x                                         
|  |  |                              2*   /   1 - --    2*a*   /   1 - --                                        
|  |  |                                  /         2          /         2                                        
|  |  \                                \/         a         \/         a                                         
|  |                                                                                                             
| /                                                                                                              
| a                                                                                                              
<                                                                                                                
|  a                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                             2                   4                     2            | 2|                   
| |  |        I*a               I*x                 I*x                 3*I*x             |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1               
| |  |        _________                3/2                 3/2             _________        2                    
| |  |       /       2        /      2\           /      2\               /       2        a                     
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          a         \     a /           \     a /          \/          a                               
| |  |                                                                                                           
| |  |                   ________                                                                                
| |  <                  /      2                                                                   dx   otherwise
| |  |                 /      x                                                                                  
| |  |           a*   /   1 - --                                                                                 
| |  |               /         2                                2                                                
| |  |             \/         a            a                   x                                                 
| |  |           ---------------- + ---------------- - ------------------              otherwise                 
| |  |                  2                   ________             ________                                        
| |  |                                     /      2             /      2                                         
| |  |                                    /      x             /      x                                          
| |  |                              2*   /   1 - --    2*a*   /   1 - --                                         
| |  |                                  /         2          /         2                                         
| |  \                                \/         a         \/         a                                          
| |                                                                                                              
|/                                                                                                               
\0
{−∫a0{−ia−1+x2a2+3ix22a−1+x2a2+ix22a(−1+x2a2)32−ix42a3(−1+x2a2)32for ∣x2∣a2>1a1−x2a22+a21−x2a2−x22a1−x2a2otherwise dxfor a<0∫0a{−ia−1+x2a2+3ix22a−1+x2a2+ix22a(−1+x2a2)32−ix42a3(−1+x2a2)32for ∣x2∣a2>1a1−x2a22+a21−x2a2−x22a1−x2a2otherwise dxotherwise\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
=
= 
/   0                                                                                                            
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|  |                                                                                                             
|  |  /                             2                   4                     2            | 2|                  
|  |  |        I*a               I*x                 I*x                 3*I*x             |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1              
|  |  |        _________                3/2                 3/2             _________        2                   
|  |  |       /       2        /      2\           /      2\               /       2        a                    
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          a         \     a /           \     a /          \/          a                              
|  |  |                                                                                                          
|  |  |                   ________                                                                               
|- |  <                  /      2                                                                   dx  for a < 0
|  |  |                 /      x                                                                                 
|  |  |           a*   /   1 - --                                                                                
|  |  |               /         2                                2                                               
|  |  |             \/         a            a                   x                                                
|  |  |           ---------------- + ---------------- - ------------------              otherwise                
|  |  |                  2                   ________             ________                                       
|  |  |                                     /      2             /      2                                        
|  |  |                                    /      x             /      x                                         
|  |  |                              2*   /   1 - --    2*a*   /   1 - --                                        
|  |  |                                  /         2          /         2                                        
|  |  \                                \/         a         \/         a                                         
|  |                                                                                                             
| /                                                                                                              
| a                                                                                                              
<                                                                                                                
|  a                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                             2                   4                     2            | 2|                   
| |  |        I*a               I*x                 I*x                 3*I*x             |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1               
| |  |        _________                3/2                 3/2             _________        2                    
| |  |       /       2        /      2\           /      2\               /       2        a                     
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*a*|-1 + --|      2*a *|-1 + --|      2*a*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          a         \     a /           \     a /          \/          a                               
| |  |                                                                                                           
| |  |                   ________                                                                                
| |  <                  /      2                                                                   dx   otherwise
| |  |                 /      x                                                                                  
| |  |           a*   /   1 - --                                                                                 
| |  |               /         2                                2                                                
| |  |             \/         a            a                   x                                                 
| |  |           ---------------- + ---------------- - ------------------              otherwise                 
| |  |                  2                   ________             ________                                        
| |  |                                     /      2             /      2                                         
| |  |                                    /      x             /      x                                          
| |  |                              2*   /   1 - --    2*a*   /   1 - --                                         
| |  |                                  /         2          /         2                                         
| |  \                                \/         a         \/         a                                          
| |                                                                                                              
|/                                                                                                               
\0
{−∫a0{−ia−1+x2a2+3ix22a−1+x2a2+ix22a(−1+x2a2)32−ix42a3(−1+x2a2)32for ∣x2∣a2>1a1−x2a22+a21−x2a2−x22a1−x2a2otherwise dxfor a<0∫0a{−ia−1+x2a2+3ix22a−1+x2a2+ix22a(−1+x2a2)32−ix42a3(−1+x2a2)32for ∣x2∣a2>1a1−x2a22+a21−x2a2−x22a1−x2a2otherwise dxotherwise\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{i a}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{2}}{2 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{a \sqrt{1 - \frac{x^{2}}{a^{2}}}}{2} + \frac{a}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
Piecewise((-Integral(Piecewise((-i*a/sqrt(-1 + x^2/a^2) + i*x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - i*x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*i*x^2/(2*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (a*sqrt(1 - x^2/a^2)/2 + a/(2*sqrt(1 - x^2/a^2)) - x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, a, 0)), a < 0), (Integral(Piecewise((-i*a/sqrt(-1 + x^2/a^2) + i*x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - i*x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*i*x^2/(2*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (a*sqrt(1 - x^2/a^2)/2 + a/(2*sqrt(1 - x^2/a^2)) - x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, 0, a)), True))

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

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