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

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] 
  a                   
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0
∫0ax2a2−x2 dx\int\limits_{0}^{a} x^{2} \sqrt{a^{2} - x^{2}}\, dx 
Integral(x^2*sqrt(a^2 - x^2), (x, 0, a))
Respuesta (Indefinida) [src] 
                            //     4      /x\                                                                            \
                            ||  I*a *acosh|-|               3                  5                     3           | 2|    |
                            ||            \a/        3*I*a*x                I*x                 I*x*a            |x |    |
                            ||- ------------- - ----------------- + ------------------- + -----------------  for |--| > 1|
                            ||        8                 _________             _________           _________      | 2|    |
                            ||                         /       2             /       2           /       2       |a |    |
  /                         ||                        /       x             /       x           /       x                |
 |                          ||                  8*   /   -1 + --    4*a*   /   -1 + --    8*   /   -1 + --               |
 |       _________          ||                      /          2          /          2        /          2               |
 |  2   /  2    2           ||                    \/          a         \/          a       \/          a                |
 | x *\/  a  - x   dx = C + |<                                                                                           |
 |                          ||     4     /x\                                                                             |
/                           ||    a *asin|-|            5                    3                   3                       |
                            ||           \a/           x                  x*a               3*a*x                        |
                            ||    ---------- - ------------------ - ---------------- + ----------------       otherwise  |
                            ||        8                  ________           ________           ________                  |
                            ||                          /      2           /      2           /      2                   |
                            ||                         /      x           /      x           /      x                    |
                            ||                 4*a*   /   1 - --    8*   /   1 - --    8*   /   1 - --                   |
                            ||                       /         2        /         2        /         2                   |
                            \\                     \/         a       \/         a       \/         a                    /
∫x2a2−x2 dx=C+{−ia4acosh⁡(xa)8+ia3x8−1+x2a2−3iax38−1+x2a2+ix54a−1+x2a2for ∣x2a2∣>1a4asin⁡(xa)8−a3x81−x2a2+3ax381−x2a2−x54a1−x2a2otherwise\int x^{2} \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{4} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{8} + \frac{i a^{3} x}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{3 i a x^{3}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{5}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{4} \operatorname{asin}{\left(\frac{x}{a} \right)}}{8} - \frac{a^{3} x}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 a x^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{5}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
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|  |  |       9*I*a*x               I*x              I*a*x             3*I*x                5*I*x             |x |                  
|  |  |- ----------------- - ----------------- - -------------- + ---------------- + -------------------  for ---- > 1              
|  |  |          _________                 3/2              3/2                3/2             _________        2                   
|  |  |         /       2         /      2\        /      2\          /      2\               /       2        a                    
|  |  |        /       x        3 |     x |        |     x |          |     x |              /       x                              
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|  |  |    \/          a          \     a /        \     a /          \     a /          \/          a                              
|- |  <                                                                                                                dx  for a < 0
|  |  |              4                  6                  2                4                  2                                    
|  |  |           5*x                  x                a*x              3*x              9*a*x                                     
|  |  |  - ------------------ - ---------------- - ------------- + --------------- + ----------------      otherwise                
|  |  |              ________                3/2             3/2               3/2           ________                               
|  |  |             /      2         /     2\        /     2\          /     2\             /      2                                
|  |  |            /      x        3 |    x |        |    x |          |    x |            /      x                                 
|  |  |    4*a*   /   1 - --    4*a *|1 - --|      8*|1 - --|      8*a*|1 - --|      8*   /   1 - --                                
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| |  /              2                 6                  2                 4                    4            | 2|                   
| |  |       9*I*a*x               I*x              I*a*x             3*I*x                5*I*x             |x |                   
| |  |- ----------------- - ----------------- - -------------- + ---------------- + -------------------  for ---- > 1               
| |  |          _________                 3/2              3/2                3/2             _________        2                    
| |  |         /       2         /      2\        /      2\          /      2\               /       2        a                     
| |  |        /       x        3 |     x |        |     x |          |     x |              /       x                               
| |  |  8*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*a*|-1 + --|      4*a*   /   -1 + --                              
| |  |      /          2         |      2|        |      2|          |      2|            /          2                              
| |  |    \/          a          \     a /        \     a /          \     a /          \/          a                               
| |  <                                                                                                                dx   otherwise
| |  |              4                  6                  2                4                  2                                     
| |  |           5*x                  x                a*x              3*x              9*a*x                                      
| |  |  - ------------------ - ---------------- - ------------- + --------------- + ----------------      otherwise                 
| |  |              ________                3/2             3/2               3/2           ________                                
| |  |             /      2         /     2\        /     2\          /     2\             /      2                                 
| |  |            /      x        3 |    x |        |    x |          |    x |            /      x                                  
| |  |    4*a*   /   1 - --    4*a *|1 - --|      8*|1 - --|      8*a*|1 - --|      8*   /   1 - --                                 
| |  |          /         2         |     2|        |     2|          |     2|          /         2                                 
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{−∫a0{−9iax28−1+x2a2−iax28(−1+x2a2)32+5ix44a−1+x2a2+3ix48a(−1+x2a2)32−ix64a3(−1+x2a2)32for ∣x2∣a2>19ax281−x2a2−ax28(1−x2a2)32−5x44a1−x2a2+3x48a(1−x2a2)32−x64a3(1−x2a2)32otherwise dxfor a<0∫0a{−9iax28−1+x2a2−iax28(−1+x2a2)32+5ix44a−1+x2a2+3ix48a(−1+x2a2)32−ix64a3(−1+x2a2)32for ∣x2∣a2>19ax281−x2a2−ax28(1−x2a2)32−5x44a1−x2a2+3x48a(1−x2a2)32−x64a3(1−x2a2)32otherwise dxotherwise\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
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|  |  /              2                 6                  2                 4                    4            | 2|                  
|  |  |       9*I*a*x               I*x              I*a*x             3*I*x                5*I*x             |x |                  
|  |  |- ----------------- - ----------------- - -------------- + ---------------- + -------------------  for ---- > 1              
|  |  |          _________                 3/2              3/2                3/2             _________        2                   
|  |  |         /       2         /      2\        /      2\          /      2\               /       2        a                    
|  |  |        /       x        3 |     x |        |     x |          |     x |              /       x                              
|  |  |  8*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*a*|-1 + --|      4*a*   /   -1 + --                             
|  |  |      /          2         |      2|        |      2|          |      2|            /          2                             
|  |  |    \/          a          \     a /        \     a /          \     a /          \/          a                              
|- |  <                                                                                                                dx  for a < 0
|  |  |              4                  6                  2                4                  2                                    
|  |  |           5*x                  x                a*x              3*x              9*a*x                                     
|  |  |  - ------------------ - ---------------- - ------------- + --------------- + ----------------      otherwise                
|  |  |              ________                3/2             3/2               3/2           ________                               
|  |  |             /      2         /     2\        /     2\          /     2\             /      2                                
|  |  |            /      x        3 |    x |        |    x |          |    x |            /      x                                 
|  |  |    4*a*   /   1 - --    4*a *|1 - --|      8*|1 - --|      8*a*|1 - --|      8*   /   1 - --                                
|  |  |          /         2         |     2|        |     2|          |     2|          /         2                                
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| |  |       9*I*a*x               I*x              I*a*x             3*I*x                5*I*x             |x |                   
| |  |- ----------------- - ----------------- - -------------- + ---------------- + -------------------  for ---- > 1               
| |  |          _________                 3/2              3/2                3/2             _________        2                    
| |  |         /       2         /      2\        /      2\          /      2\               /       2        a                     
| |  |        /       x        3 |     x |        |     x |          |     x |              /       x                               
| |  |  8*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*a*|-1 + --|      4*a*   /   -1 + --                              
| |  |      /          2         |      2|        |      2|          |      2|            /          2                              
| |  |    \/          a          \     a /        \     a /          \     a /          \/          a                               
| |  <                                                                                                                dx   otherwise
| |  |              4                  6                  2                4                  2                                     
| |  |           5*x                  x                a*x              3*x              9*a*x                                      
| |  |  - ------------------ - ---------------- - ------------- + --------------- + ----------------      otherwise                 
| |  |              ________                3/2             3/2               3/2           ________                                
| |  |             /      2         /     2\        /     2\          /     2\             /      2                                 
| |  |            /      x        3 |    x |        |    x |          |    x |            /      x                                  
| |  |    4*a*   /   1 - --    4*a *|1 - --|      8*|1 - --|      8*a*|1 - --|      8*   /   1 - --                                 
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\0
{−∫a0{−9iax28−1+x2a2−iax28(−1+x2a2)32+5ix44a−1+x2a2+3ix48a(−1+x2a2)32−ix64a3(−1+x2a2)32for ∣x2∣a2>19ax281−x2a2−ax28(1−x2a2)32−5x44a1−x2a2+3x48a(1−x2a2)32−x64a3(1−x2a2)32otherwise dxfor a<0∫0a{−9iax28−1+x2a2−iax28(−1+x2a2)32+5ix44a−1+x2a2+3ix48a(−1+x2a2)32−ix64a3(−1+x2a2)32for ∣x2∣a2>19ax281−x2a2−ax28(1−x2a2)32−5x44a1−x2a2+3x48a(1−x2a2)32−x64a3(1−x2a2)32otherwise dxotherwise\begin{cases} - \int\limits_{a}^{0} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: a < 0 \\\int\limits_{0}^{a} \begin{cases} - \frac{9 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a x^{2}}{8 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{3 i x^{4}}{8 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{6}}{4 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{\left|{x^{2}}\right|}{a^{2}} > 1 \\\frac{9 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a x^{2}}{8 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{3 x^{4}}{8 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{6}}{4 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
Piecewise((-Integral(Piecewise((-9*i*a*x^2/(8*sqrt(-1 + x^2/a^2)) - i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) - i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 3*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) + 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (-5*x^4/(4*a*sqrt(1 - x^2/a^2)) - x^6/(4*a^3*(1 - x^2/a^2)^(3/2)) - a*x^2/(8*(1 - x^2/a^2)^(3/2)) + 3*x^4/(8*a*(1 - x^2/a^2)^(3/2)) + 9*a*x^2/(8*sqrt(1 - x^2/a^2)), True)), (x, a, 0)), a < 0), (Integral(Piecewise((-9*i*a*x^2/(8*sqrt(-1 + x^2/a^2)) - i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) - i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 3*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) + 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)), |x^2|/a^2 > 1), (-5*x^4/(4*a*sqrt(1 - x^2/a^2)) - x^6/(4*a^3*(1 - x^2/a^2)^(3/2)) - a*x^2/(8*(1 - x^2/a^2)^(3/2)) + 3*x^4/(8*a*(1 - x^2/a^2)^(3/2)) + 9*a*x^2/(8*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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