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Resolución de integrales/ 2-y^2/ (r^2-y^2)^(3/2)

Integral de (r^2-y^2)^(3/2) dy

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
     0                   
     /                   
    |                    
    |              3/2   
    |     / 2    2\      
    |     \r  - y /    dy
    |                    
   /                     
     ___                 
-r*\/ 2                  
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    2
∫−2r20(r2−y2)32 dy\int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \left(r^{2} - y^{2}\right)^{\frac{3}{2}}\, dy 
Integral((r^2 - y^2)^(3/2), (y, -r*sqrt(2)/2, 0))
Solución detallada

  1. Vuelva a escribir el integrando:

    (r2−y2)32=r2r2−y2−y2r2−y2\left(r^{2} - y^{2}\right)^{\frac{3}{2}} = r^{2} \sqrt{r^{2} - y^{2}} - y^{2} \sqrt{r^{2} - y^{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:

      ∫r2r2−y2 dy=r2∫r2−y2 dy\int r^{2} \sqrt{r^{2} - y^{2}}\, dy = r^{2} \int \sqrt{r^{2} - y^{2}}\, dy

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

        Pero la integral

        {−ir2acosh⁡(yr)2−iry2−1+y2r2+iy32r−1+y2r2for ∣y2r2∣>1r2asin⁡(yr)2+ry1−y2r22otherwese\begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwese} \end{cases}

      Por lo tanto, el resultado es: r2({−ir2acosh⁡(yr)2−iry2−1+y2r2+iy32r−1+y2r2for ∣y2r2∣>1r2asin⁡(yr)2+ry1−y2r22otherwese)r^{2} \left(\begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{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:

      ∫(−y2r2−y2) dy=−∫y2r2−y2 dy\int \left(- y^{2} \sqrt{r^{2} - y^{2}}\right)\, dy = - \int y^{2} \sqrt{r^{2} - y^{2}}\, dy

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

        Pero la integral

        {−ir4acosh⁡(yr)8+ir3y8−1+y2r2−3iry38−1+y2r2+iy54r−1+y2r2for ∣y2r2∣>1r4asin⁡(yr)8−r3y81−y2r2+3ry381−y2r2−y54r1−y2r2otherwese\begin{cases} - \frac{i r^{4} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{8} + \frac{i r^{3} y}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{3 i r y^{3}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{5}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{4} \operatorname{asin}{\left(\frac{y}{r} \right)}}{8} - \frac{r^{3} y}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{3 r y^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{5}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwese} \end{cases}

      Por lo tanto, el resultado es: −{−ir4acosh⁡(yr)8+ir3y8−1+y2r2−3iry38−1+y2r2+iy54r−1+y2r2for ∣y2r2∣>1r4asin⁡(yr)8−r3y81−y2r2+3ry381−y2r2−y54r1−y2r2otherwese- \begin{cases} - \frac{i r^{4} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{8} + \frac{i r^{3} y}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{3 i r y^{3}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{5}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{4} \operatorname{asin}{\left(\frac{y}{r} \right)}}{8} - \frac{r^{3} y}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{3 r y^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{5}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwese} \end{cases}

    El resultado es: r2({−ir2acosh⁡(yr)2−iry2−1+y2r2+iy32r−1+y2r2for ∣y2r2∣>1r2asin⁡(yr)2+ry1−y2r22otherwese)−{−ir4acosh⁡(yr)8+ir3y8−1+y2r2−3iry38−1+y2r2+iy54r−1+y2r2for ∣y2r2∣>1r4asin⁡(yr)8−r3y81−y2r2+3ry381−y2r2−y54r1−y2r2otherweser^{2} \left(\begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwese} \end{cases}\right) - \begin{cases} - \frac{i r^{4} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{8} + \frac{i r^{3} y}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{3 i r y^{3}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{5}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{4} \operatorname{asin}{\left(\frac{y}{r} \right)}}{8} - \frac{r^{3} y}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{3 r y^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{5}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwese} \end{cases}

  3. Ahora simplificar:

    {ir(−3r3acosh⁡(yr)+5r2y−r2+y2r2−2y3−r2+y2r2)8for ∣y2r2∣>1r(3r3asin⁡(yr)+5r2y1−y2r2−2y31−y2r2)8otherwese\begin{cases} \frac{i r \left(- 3 r^{3} \operatorname{acosh}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}} - 2 y^{3} \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}}\right)}{8} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r \left(3 r^{3} \operatorname{asin}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{1 - \frac{y^{2}}{r^{2}}} - 2 y^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}\right)}{8} & \text{otherwese} \end{cases}

  4. Añadimos la constante de integración:

    {ir(−3r3acosh⁡(yr)+5r2y−r2+y2r2−2y3−r2+y2r2)8for ∣y2r2∣>1r(3r3asin⁡(yr)+5r2y1−y2r2−2y31−y2r2)8otherwese+constant\begin{cases} \frac{i r \left(- 3 r^{3} \operatorname{acosh}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}} - 2 y^{3} \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}}\right)}{8} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r \left(3 r^{3} \operatorname{asin}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{1 - \frac{y^{2}}{r^{2}}} - 2 y^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}\right)}{8} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta:

{ir(−3r3acosh⁡(yr)+5r2y−r2+y2r2−2y3−r2+y2r2)8for ∣y2r2∣>1r(3r3asin⁡(yr)+5r2y1−y2r2−2y31−y2r2)8otherwese+constant\begin{cases} \frac{i r \left(- 3 r^{3} \operatorname{acosh}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}} - 2 y^{3} \sqrt{\frac{- r^{2} + y^{2}}{r^{2}}}\right)}{8} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r \left(3 r^{3} \operatorname{asin}{\left(\frac{y}{r} \right)} + 5 r^{2} y \sqrt{1 - \frac{y^{2}}{r^{2}}} - 2 y^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}\right)}{8} & \text{otherwese} \end{cases}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
                         //     4      /y\                                                                            \                                                                                
                         ||  I*r *acosh|-|               3                  5                     3           | 2|    |      //     2      /y\                                                        \
                         ||            \r/        3*I*r*y                I*y                 I*y*r            |y |    |      ||  I*r *acosh|-|              3                                 | 2|    |
                         ||- ------------- - ----------------- + ------------------- + -----------------  for |--| > 1|      ||            \r/           I*y                 I*r*y            |y |    |
                         ||        8                 _________             _________           _________      | 2|    |      ||- ------------- + ------------------- - -----------------  for |--| > 1|
                         ||                         /       2             /       2           /       2       |r |    |      ||        2                   _________           _________      | 2|    |
  /                      ||                        /       y             /       y           /       y                |      ||                           /       2           /       2       |r |    |
 |                       ||                  8*   /   -1 + --    4*r*   /   -1 + --    8*   /   -1 + --               |      ||                          /       y           /       y                |
 |          3/2          ||                      /          2          /          2        /          2               |      ||                  2*r*   /   -1 + --    2*   /   -1 + --               |
 | / 2    2\             ||                    \/          r         \/          r       \/          r                |    2 ||                        /          2        /          2               |
 | \r  - y /    dy = C - |<                                                                                           | + r *|<                      \/          r       \/          r                |
 |                       ||     4     /y\                                                                             |      ||                                                                       |
/                        ||    r *asin|-|            5                    3                   3                       |      ||                                    ________                           |
                         ||           \r/           y                  y*r               3*r*y                        |      ||                                   /      2                            |
                         ||    ---------- - ------------------ - ---------------- + ----------------       otherwise  |      ||                                  /      y                             |
                         ||        8                  ________           ________           ________                  |      ||              2     /y\   r*y*   /   1 - --                            |
                         ||                          /      2           /      2           /      2                   |      ||             r *asin|-|         /         2                            |
                         ||                         /      y           /      y           /      y                    |      ||                    \r/       \/         r                             |
                         ||                 4*r*   /   1 - --    8*   /   1 - --    8*   /   1 - --                   |      ||             ---------- + ------------------                otherwise  |
                         ||                       /         2        /         2        /         2                   |      \\                 2                2                                    /
                         \\                     \/         r       \/         r       \/         r                    /
∫(r2−y2)32 dy=C+r2({−ir2acosh⁡(yr)2−iry2−1+y2r2+iy32r−1+y2r2for ∣y2r2∣>1r2asin⁡(yr)2+ry1−y2r22otherwise)−{−ir4acosh⁡(yr)8+ir3y8−1+y2r2−3iry38−1+y2r2+iy54r−1+y2r2for ∣y2r2∣>1r4asin⁡(yr)8−r3y81−y2r2+3ry381−y2r2−y54r1−y2r2otherwise\int \left(r^{2} - y^{2}\right)^{\frac{3}{2}}\, dy = C + r^{2} \left(\begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{2} - \frac{i r y}{2 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{3}}{2 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{y}{r} \right)}}{2} + \frac{r y \sqrt{1 - \frac{y^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}\right) - \begin{cases} - \frac{i r^{4} \operatorname{acosh}{\left(\frac{y}{r} \right)}}{8} + \frac{i r^{3} y}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{3 i r y^{3}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{i y^{5}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{r^{4} \operatorname{asin}{\left(\frac{y}{r} \right)}}{8} - \frac{r^{3} y}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{3 r y^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{y^{5}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
/     0                                                                                                                                                           
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|    |     /           3                  4                    4                   6                   2                  2          | 2|                         
|    |     |        I*r              7*I*y                5*I*y                 I*y             5*I*r*y           21*I*r*y           |y |                         
|    |     |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1                     
|    |     |        _________                3/2             _________                 3/2              3/2           _________      | 2|                         
|    |     |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |r |                         
|    |     |      /       y         |     y |              /       y        3 |     y |        |     y |            /       y                                     
|    |     |     /   -1 + --    8*r*|-1 + --|      4*r*   /   -1 + --    4*r *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                                    
|    |     |    /          2        |      2|            /          2         |      2|        |      2|          /          2                                    
|    |     |  \/          r         \     r /          \/          r          \     r /        \     r /        \/          r                                     
|    |     |                                                                                                                                               ___    
|    |     |                                    ________                                   ________                                                    r*\/ 2     
|    |     <                                   /      2                                   /      2                                            dy   for ------- > 0
|    |     |                            3     /      y                             2     /      y                                                         2       
|    |     |                         5*r *   /   1 - --                       3*r*y *   /   1 - --                                                                
|    |     |               3                /         2              2                 /         2             4                                                  
|    |     |            3*r               \/         r          5*r*y                \/         r             y                                                   
|    |     |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                       
|    |     |              ________            8                    ________             4                       ________                                          
|    |     |             /      2                                 /      2                                     /      2                                           
|    |     |            /      y                                 /      y                                     /      y                                            
|    |     |      8*   /   1 - --                          8*   /   1 - --                            4*r*   /   1 - --                                           
|    |     |          /         2                              /         2                                  /         2                                           
|    |     \        \/         r                             \/         r                                 \/         r                                            
|    |                                                                                                                                                            
|   /                                                                                                                                                             
|     ___                                                                                                                                                         
|-r*\/ 2                                                                                                                                                          
|---------                                                                                                                                                        
|    2                                                                                                                                                            
<                                                                                                                                                                 
|       ___                                                                                                                                                       
|  -r*\/ 2                                                                                                                                                        
|  ---------                                                                                                                                                      
|      2                                                                                                                                                          
|      /                                                                                                                                                          
|     |                                                                                                                                                           
|     |     /           3                  4                    4                   6                   2                  2          | 2|                        
|     |     |        I*r              7*I*y                5*I*y                 I*y             5*I*r*y           21*I*r*y           |y |                        
|     |     |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1                    
|     |     |        _________                3/2             _________                 3/2              3/2           _________      | 2|                        
|     |     |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |r |                        
|     |     |      /       y         |     y |              /       y        3 |     y |        |     y |            /       y                                    
|     |     |     /   -1 + --    8*r*|-1 + --|      4*r*   /   -1 + --    4*r *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                                   
|     |     |    /          2        |      2|            /          2         |      2|        |      2|          /          2                                   
|     |     |  \/          r         \     r /          \/          r          \     r /        \     r /        \/          r                                    
|     |     |                                                                                                                                                     
|     |     |                                    ________                                   ________                                                              
|-    |     <                                   /      2                                   /      2                                            dy     otherwise   
|     |     |                            3     /      y                             2     /      y                                                                
|     |     |                         5*r *   /   1 - --                       3*r*y *   /   1 - --                                                               
|     |     |               3                /         2              2                 /         2             4                                                 
|     |     |            3*r               \/         r          5*r*y                \/         r             y                                                  
|     |     |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                      
|     |     |              ________            8                    ________             4                       ________                                         
|     |     |             /      2                                 /      2                                     /      2                                          
|     |     |            /      y                                 /      y                                     /      y                                           
|     |     |      8*   /   1 - --                          8*   /   1 - --                            4*r*   /   1 - --                                          
|     |     |          /         2                              /         2                                  /         2                                          
|     |     \        \/         r                             \/         r                                 \/         r                                           
|     |                                                                                                                                                           
|    /                                                                                                                                                            
\    0
{∫−2r20{−ir3−1+y2r2+21iry28−1+y2r2+5iry28(−1+y2r2)32−5iy44r−1+y2r2−7iy48r(−1+y2r2)32+iy64r3(−1+y2r2)32for ∣y2r2∣>15r31−y2r28+3r381−y2r2−3ry21−y2r24−5ry281−y2r2+y44r1−y2r2otherwise dyfor 2r2>0−∫0−2r2{−ir3−1+y2r2+21iry28−1+y2r2+5iry28(−1+y2r2)32−5iy44r−1+y2r2−7iy48r(−1+y2r2)32+iy64r3(−1+y2r2)32for ∣y2r2∣>15r31−y2r28+3r381−y2r2−3ry21−y2r24−5ry281−y2r2+y44r1−y2r2otherwise dyotherwise\begin{cases} \int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \frac{\sqrt{2} r}{2} > 0 \\- \int\limits_{0}^{- \frac{\sqrt{2} r}{2}} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases} 
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/     0                                                                                                                                                           
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|    |     /           3                  4                    4                   6                   2                  2          | 2|                         
|    |     |        I*r              7*I*y                5*I*y                 I*y             5*I*r*y           21*I*r*y           |y |                         
|    |     |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1                     
|    |     |        _________                3/2             _________                 3/2              3/2           _________      | 2|                         
|    |     |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |r |                         
|    |     |      /       y         |     y |              /       y        3 |     y |        |     y |            /       y                                     
|    |     |     /   -1 + --    8*r*|-1 + --|      4*r*   /   -1 + --    4*r *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                                    
|    |     |    /          2        |      2|            /          2         |      2|        |      2|          /          2                                    
|    |     |  \/          r         \     r /          \/          r          \     r /        \     r /        \/          r                                     
|    |     |                                                                                                                                               ___    
|    |     |                                    ________                                   ________                                                    r*\/ 2     
|    |     <                                   /      2                                   /      2                                            dy   for ------- > 0
|    |     |                            3     /      y                             2     /      y                                                         2       
|    |     |                         5*r *   /   1 - --                       3*r*y *   /   1 - --                                                                
|    |     |               3                /         2              2                 /         2             4                                                  
|    |     |            3*r               \/         r          5*r*y                \/         r             y                                                   
|    |     |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                       
|    |     |              ________            8                    ________             4                       ________                                          
|    |     |             /      2                                 /      2                                     /      2                                           
|    |     |            /      y                                 /      y                                     /      y                                            
|    |     |      8*   /   1 - --                          8*   /   1 - --                            4*r*   /   1 - --                                           
|    |     |          /         2                              /         2                                  /         2                                           
|    |     \        \/         r                             \/         r                                 \/         r                                            
|    |                                                                                                                                                            
|   /                                                                                                                                                             
|     ___                                                                                                                                                         
|-r*\/ 2                                                                                                                                                          
|---------                                                                                                                                                        
|    2                                                                                                                                                            
<                                                                                                                                                                 
|       ___                                                                                                                                                       
|  -r*\/ 2                                                                                                                                                        
|  ---------                                                                                                                                                      
|      2                                                                                                                                                          
|      /                                                                                                                                                          
|     |                                                                                                                                                           
|     |     /           3                  4                    4                   6                   2                  2          | 2|                        
|     |     |        I*r              7*I*y                5*I*y                 I*y             5*I*r*y           21*I*r*y           |y |                        
|     |     |- --------------- - ---------------- - ------------------- + ----------------- + -------------- + -----------------  for |--| > 1                    
|     |     |        _________                3/2             _________                 3/2              3/2           _________      | 2|                        
|     |     |       /       2        /      2\               /       2         /      2\        /      2\             /       2       |r |                        
|     |     |      /       y         |     y |              /       y        3 |     y |        |     y |            /       y                                    
|     |     |     /   -1 + --    8*r*|-1 + --|      4*r*   /   -1 + --    4*r *|-1 + --|      8*|-1 + --|      8*   /   -1 + --                                   
|     |     |    /          2        |      2|            /          2         |      2|        |      2|          /          2                                   
|     |     |  \/          r         \     r /          \/          r          \     r /        \     r /        \/          r                                    
|     |     |                                                                                                                                                     
|     |     |                                    ________                                   ________                                                              
|-    |     <                                   /      2                                   /      2                                            dy     otherwise   
|     |     |                            3     /      y                             2     /      y                                                                
|     |     |                         5*r *   /   1 - --                       3*r*y *   /   1 - --                                                               
|     |     |               3                /         2              2                 /         2             4                                                 
|     |     |            3*r               \/         r          5*r*y                \/         r             y                                                  
|     |     |      ---------------- + ------------------- - ---------------- - --------------------- + ------------------          otherwise                      
|     |     |              ________            8                    ________             4                       ________                                         
|     |     |             /      2                                 /      2                                     /      2                                          
|     |     |            /      y                                 /      y                                     /      y                                           
|     |     |      8*   /   1 - --                          8*   /   1 - --                            4*r*   /   1 - --                                          
|     |     |          /         2                              /         2                                  /         2                                          
|     |     \        \/         r                             \/         r                                 \/         r                                           
|     |                                                                                                                                                           
|    /                                                                                                                                                            
\    0
{∫−2r20{−ir3−1+y2r2+21iry28−1+y2r2+5iry28(−1+y2r2)32−5iy44r−1+y2r2−7iy48r(−1+y2r2)32+iy64r3(−1+y2r2)32for ∣y2r2∣>15r31−y2r28+3r381−y2r2−3ry21−y2r24−5ry281−y2r2+y44r1−y2r2otherwise dyfor 2r2>0−∫0−2r2{−ir3−1+y2r2+21iry28−1+y2r2+5iry28(−1+y2r2)32−5iy44r−1+y2r2−7iy48r(−1+y2r2)32+iy64r3(−1+y2r2)32for ∣y2r2∣>15r31−y2r28+3r381−y2r2−3ry21−y2r24−5ry281−y2r2+y44r1−y2r2otherwise dyotherwise\begin{cases} \int\limits_{- \frac{\sqrt{2} r}{2}}^{0} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{for}\: \frac{\sqrt{2} r}{2} > 0 \\- \int\limits_{0}^{- \frac{\sqrt{2} r}{2}} \begin{cases} - \frac{i r^{3}}{\sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{21 i r y^{2}}{8 \sqrt{-1 + \frac{y^{2}}{r^{2}}}} + \frac{5 i r y^{2}}{8 \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} - \frac{5 i y^{4}}{4 r \sqrt{-1 + \frac{y^{2}}{r^{2}}}} - \frac{7 i y^{4}}{8 r \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} + \frac{i y^{6}}{4 r^{3} \left(-1 + \frac{y^{2}}{r^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \left|{\frac{y^{2}}{r^{2}}}\right| > 1 \\\frac{5 r^{3} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{8} + \frac{3 r^{3}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} - \frac{3 r y^{2} \sqrt{1 - \frac{y^{2}}{r^{2}}}}{4} - \frac{5 r y^{2}}{8 \sqrt{1 - \frac{y^{2}}{r^{2}}}} + \frac{y^{4}}{4 r \sqrt{1 - \frac{y^{2}}{r^{2}}}} & \text{otherwise} \end{cases}\, dy & \text{otherwise} \end{cases} 
Piecewise((Integral(Piecewise((-i*r^3/sqrt(-1 + y^2/r^2) - 7*i*y^4/(8*r*(-1 + y^2/r^2)^(3/2)) - 5*i*y^4/(4*r*sqrt(-1 + y^2/r^2)) + i*y^6/(4*r^3*(-1 + y^2/r^2)^(3/2)) + 5*i*r*y^2/(8*(-1 + y^2/r^2)^(3/2)) + 21*i*r*y^2/(8*sqrt(-1 + y^2/r^2)), |y^2/r^2| > 1), (3*r^3/(8*sqrt(1 - y^2/r^2)) + 5*r^3*sqrt(1 - y^2/r^2)/8 - 5*r*y^2/(8*sqrt(1 - y^2/r^2)) - 3*r*y^2*sqrt(1 - y^2/r^2)/4 + y^4/(4*r*sqrt(1 - y^2/r^2)), True)), (y, -r*sqrt(2)/2, 0)), r*sqrt(2)/2 > 0), (-Integral(Piecewise((-i*r^3/sqrt(-1 + y^2/r^2) - 7*i*y^4/(8*r*(-1 + y^2/r^2)^(3/2)) - 5*i*y^4/(4*r*sqrt(-1 + y^2/r^2)) + i*y^6/(4*r^3*(-1 + y^2/r^2)^(3/2)) + 5*i*r*y^2/(8*(-1 + y^2/r^2)^(3/2)) + 21*i*r*y^2/(8*sqrt(-1 + y^2/r^2)), |y^2/r^2| > 1), (3*r^3/(8*sqrt(1 - y^2/r^2)) + 5*r^3*sqrt(1 - y^2/r^2)/8 - 5*r*y^2/(8*sqrt(1 - y^2/r^2)) - 3*r*y^2*sqrt(1 - y^2/r^2)/4 + y^4/(4*r*sqrt(1 - y^2/r^2)), True)), (y, 0, -r*sqrt(2)/2)), True))

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

    © Sr Examen – Калькулятор