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Resolución de integrales/ 2-x^2/ (a^2-x^2)^1.5

Integral de (a^2-x^2)^1.5 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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0
∫0r(a2−x2)32 dx\int\limits_{0}^{r} \left(a^{2} - x^{2}\right)^{\frac{3}{2}}\, dx 
Integral((a^2 - x^2)^(3/2), (x, 0, r))
Solución detallada

  1. Vuelva a escribir el integrando:

    (a2−x2)32=a2a2−x2−x2a2−x2\left(a^{2} - x^{2}\right)^{\frac{3}{2}} = a^{2} \sqrt{a^{2} - x^{2}} - x^{2} \sqrt{a^{2} - x^{2}}

  2. Integramos término a término:

    1. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫a2a2−x2 dx=a2∫a2−x2 dx\int a^{2} \sqrt{a^{2} - x^{2}}\, dx = a^{2} \int \sqrt{a^{2} - x^{2}}\, dx

      1. No puedo encontrar los pasos en la búsqueda de esta integral.

        Pero la integral

        {−ia2acosh⁡(xa)2−iax2−1+x2a2+ix32a−1+x2a2for ∣x2a2∣>1a2asin⁡(xa)2+ax1−x2a22otherwese\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{otherwese} \end{cases}

      Por lo tanto, el resultado es: a2({−ia2acosh⁡(xa)2−iax2−1+x2a2+ix32a−1+x2a2for ∣x2a2∣>1a2asin⁡(xa)2+ax1−x2a22otherwese)a^{2} \left(\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{otherwese} \end{cases}\right)

    1. La integral del producto de una función por una constante es la constante por la integral de esta función:

      ∫(−x2a2−x2) dx=−∫x2a2−x2 dx\int \left(- x^{2} \sqrt{a^{2} - x^{2}}\right)\, dx = - \int x^{2} \sqrt{a^{2} - x^{2}}\, dx

      1. No puedo encontrar los pasos en la búsqueda de esta integral.

        Pero la integral

        {−ia4acosh⁡(xa)8+ia3x8−1+x2a2−3iax38−1+x2a2+ix54a−1+x2a2for ∣x2a2∣>1a4asin⁡(xa)8−a3x81−x2a2+3ax381−x2a2−x54a1−x2a2otherwese\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{otherwese} \end{cases}

      Por lo tanto, el resultado es: −{−ia4acosh⁡(xa)8+ia3x8−1+x2a2−3iax38−1+x2a2+ix54a−1+x2a2for ∣x2a2∣>1a4asin⁡(xa)8−a3x81−x2a2+3ax381−x2a2−x54a1−x2a2otherwese- \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{otherwese} \end{cases}

    El resultado es: a2({−ia2acosh⁡(xa)2−iax2−1+x2a2+ix32a−1+x2a2for ∣x2a2∣>1a2asin⁡(xa)2+ax1−x2a22otherwese)−{−ia4acosh⁡(xa)8+ia3x8−1+x2a2−3iax38−1+x2a2+ix54a−1+x2a2for ∣x2a2∣>1a4asin⁡(xa)8−a3x81−x2a2+3ax381−x2a2−x54a1−x2a2otherwesea^{2} \left(\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{otherwese} \end{cases}\right) - \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{otherwese} \end{cases}

  3. Ahora simplificar:

    {ia(−3a3acosh⁡(xa)+5a2x−a2+x2a2−2x3−a2+x2a2)8for ∣x2a2∣>1a(3a3asin⁡(xa)+5a2x1−x2a2−2x31−x2a2)8otherwese\begin{cases} \frac{i a \left(- 3 a^{3} \operatorname{acosh}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}} - 2 x^{3} \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}\right)}{8} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a \left(3 a^{3} \operatorname{asin}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{1 - \frac{x^{2}}{a^{2}}} - 2 x^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}\right)}{8} & \text{otherwese} \end{cases}

  4. Añadimos la constante de integración:

    {ia(−3a3acosh⁡(xa)+5a2x−a2+x2a2−2x3−a2+x2a2)8for ∣x2a2∣>1a(3a3asin⁡(xa)+5a2x1−x2a2−2x31−x2a2)8otherwese+constant\begin{cases} \frac{i a \left(- 3 a^{3} \operatorname{acosh}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}} - 2 x^{3} \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}\right)}{8} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a \left(3 a^{3} \operatorname{asin}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{1 - \frac{x^{2}}{a^{2}}} - 2 x^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}\right)}{8} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{ia(−3a3acosh⁡(xa)+5a2x−a2+x2a2−2x3−a2+x2a2)8for ∣x2a2∣>1a(3a3asin⁡(xa)+5a2x1−x2a2−2x31−x2a2)8otherwese+constant\begin{cases} \frac{i a \left(- 3 a^{3} \operatorname{acosh}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}} - 2 x^{3} \sqrt{\frac{- a^{2} + x^{2}}{a^{2}}}\right)}{8} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a \left(3 a^{3} \operatorname{asin}{\left(\frac{x}{a} \right)} + 5 a^{2} x \sqrt{1 - \frac{x^{2}}{a^{2}}} - 2 x^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}\right)}{8} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                         //     4      /x\                                                                            \                                                                                
                         ||  I*a *acosh|-|               3                  5                     3           | 2|    |      //     2      /x\                                                        \
                         ||            \a/        3*I*a*x                I*x                 I*x*a            |x |    |      ||  I*a *acosh|-|              3                                 | 2|    |
                         ||- ------------- - ----------------- + ------------------- + -----------------  for |--| > 1|      ||            \a/           I*x                 I*a*x            |x |    |
                         ||        8                 _________             _________           _________      | 2|    |      ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||                         /       2             /       2           /       2       |a |    |      ||        2                   _________           _________      | 2|    |
  /                      ||                        /       x             /       x           /       x                |      ||                           /       2           /       2       |a |    |
 |                       ||                  8*   /   -1 + --    4*a*   /   -1 + --    8*   /   -1 + --               |      ||                          /       x           /       x                |
 |          3/2          ||                      /          2          /          2        /          2               |      ||                  2*a*   /   -1 + --    2*   /   -1 + --               |
 | / 2    2\             ||                    \/          a         \/          a       \/          a                |    2 ||                        /          2        /          2               |
 | \a  - x /    dx = C - |<                                                                                           | + a *|<                      \/          a       \/          a                |
 |                       ||     4     /x\                                                                             |      ||                                                                       |
/                        ||    a *asin|-|            5                    3                   3                       |      ||                                    ________                           |
                         ||           \a/           x                  x*a               3*a*x                        |      ||                                   /      2                            |
                         ||    ---------- - ------------------ - ---------------- + ----------------       otherwise  |      ||                                  /      x                             |
                         ||        8                  ________           ________           ________                  |      ||              2     /x\   a*x*   /   1 - --                            |
                         ||                          /      2           /      2           /      2                   |      ||             a *asin|-|         /         2                            |
                         ||                         /      x           /      x           /      x                    |      ||                    \a/       \/         a                             |
                         ||                 4*a*   /   1 - --    8*   /   1 - --    8*   /   1 - --                   |      ||             ---------- + ------------------                otherwise  |
                         ||                       /         2        /         2        /         2                   |      \\                 2                2                                    /
                         \\                     \/         a       \/         a       \/         a                    /
∫(a2−x2)32 dx=C+a2({−ia2acosh⁡(xa)2−iax2−1+x2a2+ix32a−1+x2a2for ∣x2a2∣>1a2asin⁡(xa)2+ax1−x2a22otherwise)−{−ia4acosh⁡(xa)8+ia3x8−1+x2a2−3iax38−1+x2a2+ix54a−1+x2a2for ∣x2a2∣>1a4asin⁡(xa)8−a3x81−x2a2+3ax381−x2a2−x54a1−x2a2otherwise\int \left(a^{2} - x^{2}\right)^{\frac{3}{2}}\, dx = C + a^{2} \left(\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}\right) - \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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|  |  /           3                  4                    4                   6                   2                  2          | 2|                  
|  |  |        I*a              7*I*x                5*I*x                 I*x             5*I*a*x           21*I*a*x           |x |                  
|  |  |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1              
|  |  |        _________                3/2             _________                 3/2              3/2           _________      | 2|                  
|  |  |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |a |                  
|  |  |      /       x         |     x |              /       x        3 |     x |        |     x |            /       x                              
|  |  |     /   -1 + --    8*a*|-1 + --|      4*a*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                             
|  |  |    /          2        |      2|            /          2         |      2|        |      2|          /          2                             
|  |  |  \/          a         \     a /          \/          a          \     a /        \     a /        \/          a                              
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|  |  |                                    ________                                   ________                                                        
|- |  <                                   /      2                                   /      2                                            dx  for r < 0
|  |  |                            3     /      x                             2     /      x                                                          
|  |  |                         5*a *   /   1 - --                       3*a*x *   /   1 - --                                                         
|  |  |               3                /         2              2                 /         2             4                                           
|  |  |            3*a               \/         a          5*a*x                \/         a             x                                            
|  |  |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                
|  |  |              ________            8                    ________             4                       ________                                   
|  |  |             /      2                                 /      2                                     /      2                                    
|  |  |            /      x                                 /      x                                     /      x                                     
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| |  /           3                  4                    4                   6                   2                  2          | 2|                   
| |  |        I*a              7*I*x                5*I*x                 I*x             5*I*a*x           21*I*a*x           |x |                   
| |  |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1               
| |  |        _________                3/2             _________                 3/2              3/2           _________      | 2|                   
| |  |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |a |                   
| |  |      /       x         |     x |              /       x        3 |     x |        |     x |            /       x                               
| |  |     /   -1 + --    8*a*|-1 + --|      4*a*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                              
| |  |    /          2        |      2|            /          2         |      2|        |      2|          /          2                              
| |  |  \/          a         \     a /          \/          a          \     a /        \     a /        \/          a                               
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| |  |                                    ________                                   ________                                                         
| |  <                                   /      2                                   /      2                                            dx   otherwise
| |  |                            3     /      x                             2     /      x                                                           
| |  |                         5*a *   /   1 - --                       3*a*x *   /   1 - --                                                          
| |  |               3                /         2              2                 /         2             4                                            
| |  |            3*a               \/         a          5*a*x                \/         a             x                                             
| |  |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                 
| |  |              ________            8                    ________             4                       ________                                    
| |  |             /      2                                 /      2                                     /      2                                     
| |  |            /      x                                 /      x                                     /      x                                      
| |  |      8*   /   1 - --                          8*   /   1 - --                            4*a*   /   1 - --                                     
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{−∫r0{−ia3−1+x2a2+21iax28−1+x2a2+5iax28(−1+x2a2)32−5ix44a−1+x2a2−7ix48a(−1+x2a2)32+ix64a3(−1+x2a2)32for ∣x2a2∣>15a31−x2a28+3a381−x2a2−3ax21−x2a24−5ax281−x2a2+x44a1−x2a2otherwise dxfor r<0∫0r{−ia3−1+x2a2+21iax28−1+x2a2+5iax28(−1+x2a2)32−5ix44a−1+x2a2−7ix48a(−1+x2a2)32+ix64a3(−1+x2a2)32for ∣x2a2∣>15a31−x2a28+3a381−x2a2−3ax21−x2a24−5ax281−x2a2+x44a1−x2a2otherwise dxotherwise\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 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{7 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}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 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{7 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}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
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|  |  |        I*a              7*I*x                5*I*x                 I*x             5*I*a*x           21*I*a*x           |x |                  
|  |  |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1              
|  |  |        _________                3/2             _________                 3/2              3/2           _________      | 2|                  
|  |  |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |a |                  
|  |  |      /       x         |     x |              /       x        3 |     x |        |     x |            /       x                              
|  |  |     /   -1 + --    8*a*|-1 + --|      4*a*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                             
|  |  |    /          2        |      2|            /          2         |      2|        |      2|          /          2                             
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|  |  |                                    ________                                   ________                                                        
|- |  <                                   /      2                                   /      2                                            dx  for r < 0
|  |  |                            3     /      x                             2     /      x                                                          
|  |  |                         5*a *   /   1 - --                       3*a*x *   /   1 - --                                                         
|  |  |               3                /         2              2                 /         2             4                                           
|  |  |            3*a               \/         a          5*a*x                \/         a             x                                            
|  |  |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                
|  |  |              ________            8                    ________             4                       ________                                   
|  |  |             /      2                                 /      2                                     /      2                                    
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| |  /           3                  4                    4                   6                   2                  2          | 2|                   
| |  |        I*a              7*I*x                5*I*x                 I*x             5*I*a*x           21*I*a*x           |x |                   
| |  |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1               
| |  |        _________                3/2             _________                 3/2              3/2           _________      | 2|                   
| |  |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |a |                   
| |  |      /       x         |     x |              /       x        3 |     x |        |     x |            /       x                               
| |  |     /   -1 + --    8*a*|-1 + --|      4*a*   /   -1 + --    4*a *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                              
| |  |    /          2        |      2|            /          2         |      2|        |      2|          /          2                              
| |  |  \/          a         \     a /          \/          a          \     a /        \     a /        \/          a                               
| |  |                                                                                                                                                
| |  |                                    ________                                   ________                                                         
| |  <                                   /      2                                   /      2                                            dx   otherwise
| |  |                            3     /      x                             2     /      x                                                           
| |  |                         5*a *   /   1 - --                       3*a*x *   /   1 - --                                                          
| |  |               3                /         2              2                 /         2             4                                            
| |  |            3*a               \/         a          5*a*x                \/         a             x                                             
| |  |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                 
| |  |              ________            8                    ________             4                       ________                                    
| |  |             /      2                                 /      2                                     /      2                                     
| |  |            /      x                                 /      x                                     /      x                                      
| |  |      8*   /   1 - --                          8*   /   1 - --                            4*a*   /   1 - --                                     
| |  |          /         2                              /         2                                  /         2                                     
| |  \        \/         a                             \/         a                                 \/         a                                      
| |                                                                                                                                                   
|/                                                                                                                                                    
\0
{−∫r0{−ia3−1+x2a2+21iax28−1+x2a2+5iax28(−1+x2a2)32−5ix44a−1+x2a2−7ix48a(−1+x2a2)32+ix64a3(−1+x2a2)32for ∣x2a2∣>15a31−x2a28+3a381−x2a2−3ax21−x2a24−5ax281−x2a2+x44a1−x2a2otherwise dxfor r<0∫0r{−ia3−1+x2a2+21iax28−1+x2a2+5iax28(−1+x2a2)32−5ix44a−1+x2a2−7ix48a(−1+x2a2)32+ix64a3(−1+x2a2)32for ∣x2a2∣>15a31−x2a28+3a381−x2a2−3ax21−x2a24−5ax281−x2a2+x44a1−x2a2otherwise dxotherwise\begin{cases} - \int\limits_{r}^{0} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 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{7 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}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r < 0 \\\int\limits_{0}^{r} \begin{cases} - \frac{i a^{3}}{\sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{21 i a x^{2}}{8 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 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{7 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}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{5 a^{3} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{8} + \frac{3 a^{3}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{3 a x^{2} \sqrt{1 - \frac{x^{2}}{a^{2}}}}{4} - \frac{5 a x^{2}}{8 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{x^{4}}{4 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases} 
Piecewise((-Integral(Piecewise((-i*a^3/sqrt(-1 + x^2/a^2) - 7*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) - 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)) + i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) + 5*i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 21*i*a*x^2/(8*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (3*a^3/(8*sqrt(1 - x^2/a^2)) + 5*a^3*sqrt(1 - x^2/a^2)/8 - 5*a*x^2/(8*sqrt(1 - x^2/a^2)) - 3*a*x^2*sqrt(1 - x^2/a^2)/4 + x^4/(4*a*sqrt(1 - x^2/a^2)), True)), (x, r, 0)), r < 0), (Integral(Piecewise((-i*a^3/sqrt(-1 + x^2/a^2) - 7*i*x^4/(8*a*(-1 + x^2/a^2)^(3/2)) - 5*i*x^4/(4*a*sqrt(-1 + x^2/a^2)) + i*x^6/(4*a^3*(-1 + x^2/a^2)^(3/2)) + 5*i*a*x^2/(8*(-1 + x^2/a^2)^(3/2)) + 21*i*a*x^2/(8*sqrt(-1 + x^2/a^2)), |x^2/a^2| > 1), (3*a^3/(8*sqrt(1 - x^2/a^2)) + 5*a^3*sqrt(1 - x^2/a^2)/8 - 5*a*x^2/(8*sqrt(1 - x^2/a^2)) - 3*a*x^2*sqrt(1 - x^2/a^2)/4 + x^4/(4*a*sqrt(1 - x^2/a^2)), True)), (x, 0, r)), True))

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

    © Sr Examen – Калькулятор