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

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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  r                
  /                
 |                 
 |     _________   
 |    /  2    2    
 |  \/  r  - x   dx
 |                 
/                  
0
∫0rr2−x2 dx\int\limits_{0}^{r} \sqrt{r^{2} - x^{2}}\, dx 
Integral(sqrt(r^2 - x^2), (x, 0, r))
Respuesta (Indefinida) [src] 
                         //     2      /x\                                                        \
                         ||  I*r *acosh|-|              3                                 | 2|    |
                         ||            \r/           I*x                 I*r*x            |x |    |
                         ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||        2                   _________           _________      | 2|    |
  /                      ||                           /       2           /       2       |r |    |
 |                       ||                          /       x           /       x                |
 |    _________          ||                  2*r*   /   -1 + --    2*   /   -1 + --               |
 |   /  2    2           ||                        /          2        /          2               |
 | \/  r  - x   dx = C + |<                      \/          r       \/          r                |
 |                       ||                                                                       |
/                        ||                                    ________                           |
                         ||                                   /      2                            |
                         ||                                  /      x                             |
                         ||              2     /x\   r*x*   /   1 - --                            |
                         ||             r *asin|-|         /         2                            |
                         ||                    \r/       \/         r                             |
                         ||             ---------- + ------------------                otherwise  |
                         \\                 2                2                                    /
∫r2−x2 dx=C+{−ir2acosh⁡(xr)2−irx2−1+x2r2+ix32r−1+x2r2for ∣x2r2∣>1r2asin⁡(xr)2+rx1−x2r22otherwise\int \sqrt{r^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{x}{r} \right)}}{2} - \frac{i r x}{2 \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{3}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{x}{r} \right)}}{2} + \frac{r x \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/   0                                                                                                            
|   /                                                                                                            
|  |                                                                                                             
|  |  /                             2                   4                     2            | 2|                  
|  |  |        I*r               I*x                 I*x                 3*I*x             |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1              
|  |  |        _________                3/2                 3/2             _________        2                   
|  |  |       /       2        /      2\           /      2\               /       2        r                    
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          r         \     r /           \     r /          \/          r                              
|  |  |                                                                                                          
|  |  |                   ________                                                                               
|- |  <                  /      2                                                                   dx  for r < 0
|  |  |                 /      x                                                                                 
|  |  |           r*   /   1 - --                                                                                
|  |  |               /         2                                2                                               
|  |  |             \/         r            r                   x                                                
|  |  |           ---------------- + ---------------- - ------------------              otherwise                
|  |  |                  2                   ________             ________                                       
|  |  |                                     /      2             /      2                                        
|  |  |                                    /      x             /      x                                         
|  |  |                              2*   /   1 - --    2*r*   /   1 - --                                        
|  |  |                                  /         2          /         2                                        
|  |  \                                \/         r         \/         r                                         
|  |                                                                                                             
| /                                                                                                              
| r                                                                                                              
<                                                                                                                
|  r                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                             2                   4                     2            | 2|                   
| |  |        I*r               I*x                 I*x                 3*I*x             |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1               
| |  |        _________                3/2                 3/2             _________        2                    
| |  |       /       2        /      2\           /      2\               /       2        r                     
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          r         \     r /           \     r /          \/          r                               
| |  |                                                                                                           
| |  |                   ________                                                                                
| |  <                  /      2                                                                   dx   otherwise
| |  |                 /      x                                                                                  
| |  |           r*   /   1 - --                                                                                 
| |  |               /         2                                2                                                
| |  |             \/         r            r                   x                                                 
| |  |           ---------------- + ---------------- - ------------------              otherwise                 
| |  |                  2                   ________             ________                                        
| |  |                                     /      2             /      2                                         
| |  |                                    /      x             /      x                                          
| |  |                              2*   /   1 - --    2*r*   /   1 - --                                         
| |  |                                  /         2          /         2                                         
| |  \                                \/         r         \/         r                                          
| |                                                                                                              
|/                                                                                                               
\0
{−∫r0{−ir−1+x2r2+3ix22r−1+x2r2+ix22r(−1+x2r2)32−ix42r3(−1+x2r2)32for ∣x2∣r2>1r1−x2r22+r21−x2r2−x22r1−x2r2otherwise dxfor r<0∫0r{−ir−1+x2r2+3ix22r−1+x2r2+ix22r(−1+x2r2)32−ix42r3(−1+x2r2)32for ∣x2∣r2>1r1−x2r22+r21−x2r2−x22r1−x2r2otherwise dxotherwise\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{3 i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{2}}{2 r \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 r^{3} \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{r^{2}} > 1 \\\frac{r \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{3 i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{2}}{2 r \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 r^{3} \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{r^{2}} > 1 \\\frac{r \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
=
= 
/   0                                                                                                            
|   /                                                                                                            
|  |                                                                                                             
|  |  /                             2                   4                     2            | 2|                  
|  |  |        I*r               I*x                 I*x                 3*I*x             |x |                  
|  |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1              
|  |  |        _________                3/2                 3/2             _________        2                   
|  |  |       /       2        /      2\           /      2\               /       2        r                    
|  |  |      /       x         |     x |         3 |     x |              /       x                              
|  |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                             
|  |  |    /          2        |      2|           |      2|            /          2                             
|  |  |  \/          r         \     r /           \     r /          \/          r                              
|  |  |                                                                                                          
|  |  |                   ________                                                                               
|- |  <                  /      2                                                                   dx  for r < 0
|  |  |                 /      x                                                                                 
|  |  |           r*   /   1 - --                                                                                
|  |  |               /         2                                2                                               
|  |  |             \/         r            r                   x                                                
|  |  |           ---------------- + ---------------- - ------------------              otherwise                
|  |  |                  2                   ________             ________                                       
|  |  |                                     /      2             /      2                                        
|  |  |                                    /      x             /      x                                         
|  |  |                              2*   /   1 - --    2*r*   /   1 - --                                        
|  |  |                                  /         2          /         2                                        
|  |  \                                \/         r         \/         r                                         
|  |                                                                                                             
| /                                                                                                              
| r                                                                                                              
<                                                                                                                
|  r                                                                                                             
|  /                                                                                                             
| |                                                                                                              
| |  /                             2                   4                     2            | 2|                   
| |  |        I*r               I*x                 I*x                 3*I*x             |x |                   
| |  |- --------------- + ---------------- - ----------------- + -------------------  for ---- > 1               
| |  |        _________                3/2                 3/2             _________        2                    
| |  |       /       2        /      2\           /      2\               /       2        r                     
| |  |      /       x         |     x |         3 |     x |              /       x                               
| |  |     /   -1 + --    2*r*|-1 + --|      2*r *|-1 + --|      2*r*   /   -1 + --                              
| |  |    /          2        |      2|           |      2|            /          2                              
| |  |  \/          r         \     r /           \     r /          \/          r                               
| |  |                                                                                                           
| |  |                   ________                                                                                
| |  <                  /      2                                                                   dx   otherwise
| |  |                 /      x                                                                                  
| |  |           r*   /   1 - --                                                                                 
| |  |               /         2                                2                                                
| |  |             \/         r            r                   x                                                 
| |  |           ---------------- + ---------------- - ------------------              otherwise                 
| |  |                  2                   ________             ________                                        
| |  |                                     /      2             /      2                                         
| |  |                                    /      x             /      x                                          
| |  |                              2*   /   1 - --    2*r*   /   1 - --                                         
| |  |                                  /         2          /         2                                         
| |  \                                \/         r         \/         r                                          
| |                                                                                                              
|/                                                                                                               
\0
{−∫r0{−ir−1+x2r2+3ix22r−1+x2r2+ix22r(−1+x2r2)32−ix42r3(−1+x2r2)32for ∣x2∣r2>1r1−x2r22+r21−x2r2−x22r1−x2r2otherwise dxfor r<0∫0r{−ir−1+x2r2+3ix22r−1+x2r2+ix22r(−1+x2r2)32−ix42r3(−1+x2r2)32for ∣x2∣r2>1r1−x2r22+r21−x2r2−x22r1−x2r2otherwise dxotherwise\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{3 i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{2}}{2 r \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 r^{3} \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{r^{2}} > 1 \\\frac{r \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i r}{\sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{3 i x^{2}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{2}}{2 r \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{4}}{2 r^{3} \left(-1 + \frac{x^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{r^{2}} > 1 \\\frac{r \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} + \frac{r}{2 \sqrt{1 - \frac{x^{2}}{r^{2}}}} - \frac{x^{2}}{2 r \sqrt{1 - \frac{x^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
Piecewise((-Integral(Piecewise((-i*r/sqrt(-1 + x^2/r^2) + i*x^2/(2*r*(-1 + x^2/r^2)^(3/2)) - i*x^4/(2*r^3*(-1 + x^2/r^2)^(3/2)) + 3*i*x^2/(2*r*sqrt(-1 + x^2/r^2)), |x^2|/r^2 > 1), (r*sqrt(1 - x^2/r^2)/2 + r/(2*sqrt(1 - x^2/r^2)) - x^2/(2*r*sqrt(1 - x^2/r^2)), True)), (x, r, 0)), r < 0), (Integral(Piecewise((-i*r/sqrt(-1 + x^2/r^2) + i*x^2/(2*r*(-1 + x^2/r^2)^(3/2)) - i*x^4/(2*r^3*(-1 + x^2/r^2)^(3/2)) + 3*i*x^2/(2*r*sqrt(-1 + x^2/r^2)), |x^2|/r^2 > 1), (r*sqrt(1 - x^2/r^2)/2 + r/(2*sqrt(1 - x^2/r^2)) - x^2/(2*r*sqrt(1 - x^2/r^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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