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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                
  /                
 |                 
 |        2        
 |       x         
 |  ------------ dx
 |     _________   
 |    /  2    2    
 |  \/  a  - x     
 |                 
/                  
0                  
$$\int\limits_{0}^{1} \frac{x^{2}}{\sqrt{a^{2} - x^{2}}}\, dx$$
Integral(x^2/sqrt(a^2 - x^2), (x, 0, 1))
Respuesta (Indefinida) [src]
                         //                                   _________                    \
                         ||                                  /       2                     |
                         ||                                 /       x                      |
                         ||          2      /x\   I*a*x*   /   -1 + --                     |
                         ||       I*a *acosh|-|           /          2             | 2|    |
                         ||                 \a/         \/          a              |x |    |
  /                      ||     - ------------- - ---------------------        for |--| > 1|
 |                       ||             2                   2                      | 2|    |
 |       2               ||                                                        |a |    |
 |      x                ||                                                                |
 | ------------ dx = C + |< 2     /x\                                                      |
 |    _________          ||a *asin|-|            3                                         |
 |   /  2    2           ||       \a/           x                  a*x                     |
 | \/  a  - x            ||---------- + ------------------ - ----------------   otherwise  |
 |                       ||    2                  ________           ________              |
/                        ||                      /      2           /      2               |
                         ||                     /      x           /      x                |
                         ||             2*a*   /   1 - --    2*   /   1 - --               |
                         ||                   /         2        /         2               |
                         ||                 \/         a       \/         a                |
                         \\                                                                /
$$\int \frac{x^{2}}{\sqrt{a^{2} - x^{2}}}\, dx = C + \begin{cases} - \frac{i a^{2} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{2} - \frac{i a x \sqrt{-1 + \frac{x^{2}}{a^{2}}}}{2} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{2} \operatorname{asin}{\left(\frac{x}{a} \right)}}{2} - \frac{a x}{2 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{3}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}$$
Respuesta [src]
  1                                                                                  
  /                                                                                  
 |                                                                                   
 |  /            _________                                                           
 |  |           /       2                                                            
 |  |          /       x                                                             
 |  |  I*a*   /   -1 + --                                                            
 |  |        /          2                                   2               2        
 |  |      \/          a            I*a                  I*x               x         
 |  |- ------------------- - ----------------- - -------------------  for ---- > 1   
 |  |           2                    _________             _________      | 2|       
 |  |                               /       2             /       2       |a |       
 |  |                              /       x             /       x                   
 |  |                        2*   /   -1 + --    2*a*   /   -1 + --                  
 |  <                            /          2          /          2                dx
 |  |                          \/          a         \/          a                   
 |  |                                                                                
 |  |            4                  2                   2                            
 |  |           x                  x                 3*x                             
 |  |    ---------------- - --------------- + ------------------       otherwise     
 |  |                 3/2               3/2             ________                     
 |  |         /     2\          /     2\               /      2                      
 |  |       3 |    x |          |    x |              /      x                       
 |  |    2*a *|1 - --|      2*a*|1 - --|      2*a*   /   1 - --                      
 |  |         |     2|          |     2|            /         2                      
 |  \         \    a /          \    a /          \/         a                       
 |                                                                                   
/                                                                                    
0                                                                                    
$$\int\limits_{0}^{1} \begin{cases} - \frac{i a \sqrt{-1 + \frac{x^{2}}{a^{2}}}}{2} - \frac{i a}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{3 x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{x^{4}}{2 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
=
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  1                                                                                  
  /                                                                                  
 |                                                                                   
 |  /            _________                                                           
 |  |           /       2                                                            
 |  |          /       x                                                             
 |  |  I*a*   /   -1 + --                                                            
 |  |        /          2                                   2               2        
 |  |      \/          a            I*a                  I*x               x         
 |  |- ------------------- - ----------------- - -------------------  for ---- > 1   
 |  |           2                    _________             _________      | 2|       
 |  |                               /       2             /       2       |a |       
 |  |                              /       x             /       x                   
 |  |                        2*   /   -1 + --    2*a*   /   -1 + --                  
 |  <                            /          2          /          2                dx
 |  |                          \/          a         \/          a                   
 |  |                                                                                
 |  |            4                  2                   2                            
 |  |           x                  x                 3*x                             
 |  |    ---------------- - --------------- + ------------------       otherwise     
 |  |                 3/2               3/2             ________                     
 |  |         /     2\          /     2\               /      2                      
 |  |       3 |    x |          |    x |              /      x                       
 |  |    2*a *|1 - --|      2*a*|1 - --|      2*a*   /   1 - --                      
 |  |         |     2|          |     2|            /         2                      
 |  \         \    a /          \    a /          \/         a                       
 |                                                                                   
/                                                                                    
0                                                                                    
$$\int\limits_{0}^{1} \begin{cases} - \frac{i a \sqrt{-1 + \frac{x^{2}}{a^{2}}}}{2} - \frac{i a}{2 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i x^{2}}{2 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{3 x^{2}}{2 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{2}}{2 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{x^{4}}{2 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i*a*sqrt(-1 + x^2/a^2)/2 - i*a/(2*sqrt(-1 + x^2/a^2)) - i*x^2/(2*a*sqrt(-1 + x^2/a^2)), x^2/|a^2| > 1), (x^4/(2*a^3*(1 - x^2/a^2)^(3/2)) - x^2/(2*a*(1 - x^2/a^2)^(3/2)) + 3*x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, 0, 1))

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