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Resolución de integrales/ 2-x^2/ x^4(a^2-x^2)^1/2

Integral de x^4(a^2-x^2)^1/2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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0
∫01x4a2−x2 dx\int\limits_{0}^{1} x^{4} \sqrt{a^{2} - x^{2}}\, dx 
Integral(x^4*sqrt(a^2 - x^2), (x, 0, 1))
Respuesta (Indefinida) [src] 
                            //     6      /x\                                                                                                   \
                            ||  I*a *acosh|-|               5                3  3                    7                     5            | 2|    |
                            ||            \a/        5*I*a*x              I*a *x                  I*x                 I*x*a             |x |    |
                            ||- ------------- - ------------------ - ------------------ + ------------------- + ------------------  for |--| > 1|
                            ||        16                 _________            _________             _________            _________      | 2|    |
                            ||                          /       2            /       2             /       2            /       2       |a |    |
  /                         ||                         /       x            /       x             /       x            /       x                |
 |                          ||                  24*   /   -1 + --    48*   /   -1 + --    6*a*   /   -1 + --    16*   /   -1 + --               |
 |       _________          ||                       /          2         /          2          /          2         /          2               |
 |  4   /  2    2           ||                     \/          a        \/          a         \/          a        \/          a                |
 | x *\/  a  - x   dx = C + |<                                                                                                                  |
 |                          ||     6     /x\                                                                                                    |
/                           ||    a *asin|-|            7                     5                3  3                    5                        |
                            ||           \a/           x                   x*a                a *x                5*a*x                         |
                            ||    ---------- - ------------------ - ----------------- + ----------------- + -----------------        otherwise  |
                            ||        16                 ________            ________            ________            ________                   |
                            ||                          /      2            /      2            /      2            /      2                    |
                            ||                         /      x            /      x            /      x            /      x                     |
                            ||                 6*a*   /   1 - --    16*   /   1 - --    48*   /   1 - --    24*   /   1 - --                    |
                            ||                       /         2         /         2         /         2         /         2                    |
                            \\                     \/         a        \/         a        \/         a        \/         a                     /
∫x4a2−x2 dx=C+{−ia6acosh⁡(xa)16+ia5x16−1+x2a2−ia3x348−1+x2a2−5iax524−1+x2a2+ix76a−1+x2a2for ∣x2a2∣>1a6asin⁡(xa)16−a5x161−x2a2+a3x3481−x2a2+5ax5241−x2a2−x76a1−x2a2otherwise\int x^{4} \sqrt{a^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i a^{6} \operatorname{acosh}{\left(\frac{x}{a} \right)}}{16} + \frac{i a^{5} x}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{3}}{48 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{5 i a x^{5}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i x^{7}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{a^{2}}}\right| > 1 \\\frac{a^{6} \operatorname{asin}{\left(\frac{x}{a} \right)}}{16} - \frac{a^{5} x}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a^{3} x^{3}}{48 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 a x^{5}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{x^{7}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} & \text{otherwise} \end{cases}
Simplificar
Respuesta [src] 
  1                                                                                                                                                              
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 |  /              4                  8                3  2               3  2                   4                  6                    6              2        
 |  |      25*I*a*x                I*x              I*a *x             I*a *x               I*a*x              5*I*x                7*I*x              x         
 |  |- ------------------ - ----------------- - --------------- - ------------------ + --------------- + ----------------- + -------------------  for ---- > 1   
 |  |           _________                 3/2               3/2            _________               3/2                 3/2             _________      | 2|       
 |  |          /       2         /      2\         /      2\              /       2       /      2\           /      2\               /       2       |a |       
 |  |         /       x        3 |     x |         |     x |             /       x        |     x |           |     x |              /       x                   
 |  |  24*   /   -1 + --    6*a *|-1 + --|      16*|-1 + --|      16*   /   -1 + --    48*|-1 + --|      24*a*|-1 + --|      6*a*   /   -1 + --                  
 |  |       /          2         |      2|         |      2|           /          2       |      2|           |      2|            /          2                  
 |  |     \/          a          \     a /         \     a /         \/          a        \     a /           \     a /          \/          a                   
 |  <                                                                                                                                                          dx
 |  |               6                  8               3  2               3  2                 4                 6                    4                          
 |  |            7*x                  x               a *x               a *x               a*x               5*x               25*a*x                           
 |  |   - ------------------ - ---------------- - -------------- + ----------------- + -------------- + ---------------- + -----------------       otherwise     
 |  |               ________                3/2              3/2            ________              3/2                3/2            ________                     
 |  |              /      2         /     2\         /     2\              /      2       /     2\           /     2\              /      2                      
 |  |             /      x        3 |    x |         |    x |             /      x        |    x |           |    x |             /      x                       
 |  |     6*a*   /   1 - --    6*a *|1 - --|      16*|1 - --|      16*   /   1 - --    48*|1 - --|      24*a*|1 - --|      24*   /   1 - --                      
 |  |           /         2         |     2|         |     2|           /         2       |     2|           |     2|           /         2                      
 |  \         \/         a          \    a /         \    a /         \/         a        \    a /           \    a /         \/         a                       
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0
∫01{−ia3x216−1+x2a2−ia3x216(−1+x2a2)32−25iax424−1+x2a2+iax448(−1+x2a2)32+7ix66a−1+x2a2+5ix624a(−1+x2a2)32−ix86a3(−1+x2a2)32for x2∣a2∣>1a3x2161−x2a2−a3x216(1−x2a2)32+25ax4241−x2a2+ax448(1−x2a2)32−7x66a1−x2a2+5x624a(1−x2a2)32−x86a3(1−x2a2)32otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i a^{3} x^{2}}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{2}}{16 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{25 i a x^{4}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i a x^{4}}{48 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i x^{6}}{24 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{8}}{6 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a^{3} x^{2}}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a^{3} x^{2}}{16 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{25 a x^{4}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a x^{4}}{48 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 x^{6}}{24 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{8}}{6 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx 
=
= 
  1                                                                                                                                                              
  /                                                                                                                                                              
 |                                                                                                                                                               
 |  /              4                  8                3  2               3  2                   4                  6                    6              2        
 |  |      25*I*a*x                I*x              I*a *x             I*a *x               I*a*x              5*I*x                7*I*x              x         
 |  |- ------------------ - ----------------- - --------------- - ------------------ + --------------- + ----------------- + -------------------  for ---- > 1   
 |  |           _________                 3/2               3/2            _________               3/2                 3/2             _________      | 2|       
 |  |          /       2         /      2\         /      2\              /       2       /      2\           /      2\               /       2       |a |       
 |  |         /       x        3 |     x |         |     x |             /       x        |     x |           |     x |              /       x                   
 |  |  24*   /   -1 + --    6*a *|-1 + --|      16*|-1 + --|      16*   /   -1 + --    48*|-1 + --|      24*a*|-1 + --|      6*a*   /   -1 + --                  
 |  |       /          2         |      2|         |      2|           /          2       |      2|           |      2|            /          2                  
 |  |     \/          a          \     a /         \     a /         \/          a        \     a /           \     a /          \/          a                   
 |  <                                                                                                                                                          dx
 |  |               6                  8               3  2               3  2                 4                 6                    4                          
 |  |            7*x                  x               a *x               a *x               a*x               5*x               25*a*x                           
 |  |   - ------------------ - ---------------- - -------------- + ----------------- + -------------- + ---------------- + -----------------       otherwise     
 |  |               ________                3/2              3/2            ________              3/2                3/2            ________                     
 |  |              /      2         /     2\         /     2\              /      2       /     2\           /     2\              /      2                      
 |  |             /      x        3 |    x |         |    x |             /      x        |    x |           |    x |             /      x                       
 |  |     6*a*   /   1 - --    6*a *|1 - --|      16*|1 - --|      16*   /   1 - --    48*|1 - --|      24*a*|1 - --|      24*   /   1 - --                      
 |  |           /         2         |     2|         |     2|           /         2       |     2|           |     2|           /         2                      
 |  \         \/         a          \    a /         \    a /         \/         a        \    a /           \    a /         \/         a                       
 |                                                                                                                                                               
/                                                                                                                                                                
0
∫01{−ia3x216−1+x2a2−ia3x216(−1+x2a2)32−25iax424−1+x2a2+iax448(−1+x2a2)32+7ix66a−1+x2a2+5ix624a(−1+x2a2)32−ix86a3(−1+x2a2)32for x2∣a2∣>1a3x2161−x2a2−a3x216(1−x2a2)32+25ax4241−x2a2+ax448(1−x2a2)32−7x66a1−x2a2+5x624a(1−x2a2)32−x86a3(1−x2a2)32otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i a^{3} x^{2}}{16 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} - \frac{i a^{3} x^{2}}{16 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{25 i a x^{4}}{24 \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{i a x^{4}}{48 \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 a \sqrt{-1 + \frac{x^{2}}{a^{2}}}} + \frac{5 i x^{6}}{24 a \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{i x^{8}}{6 a^{3} \left(-1 + \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{for}\: \frac{x^{2}}{\left|{a^{2}}\right|} > 1 \\\frac{a^{3} x^{2}}{16 \sqrt{1 - \frac{x^{2}}{a^{2}}}} - \frac{a^{3} x^{2}}{16 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} + \frac{25 a x^{4}}{24 \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{a x^{4}}{48 \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 a \sqrt{1 - \frac{x^{2}}{a^{2}}}} + \frac{5 x^{6}}{24 a \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} - \frac{x^{8}}{6 a^{3} \left(1 - \frac{x^{2}}{a^{2}}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-25*i*a*x^4/(24*sqrt(-1 + x^2/a^2)) - i*x^8/(6*a^3*(-1 + x^2/a^2)^(3/2)) - i*a^3*x^2/(16*(-1 + x^2/a^2)^(3/2)) - i*a^3*x^2/(16*sqrt(-1 + x^2/a^2)) + i*a*x^4/(48*(-1 + x^2/a^2)^(3/2)) + 5*i*x^6/(24*a*(-1 + x^2/a^2)^(3/2)) + 7*i*x^6/(6*a*sqrt(-1 + x^2/a^2)), x^2/|a^2| > 1), (-7*x^6/(6*a*sqrt(1 - x^2/a^2)) - x^8/(6*a^3*(1 - x^2/a^2)^(3/2)) - a^3*x^2/(16*(1 - x^2/a^2)^(3/2)) + a^3*x^2/(16*sqrt(1 - x^2/a^2)) + a*x^4/(48*(1 - x^2/a^2)^(3/2)) + 5*x^6/(24*a*(1 - x^2/a^2)^(3/2)) + 25*a*x^4/(24*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.

    © Sr Examen – Калькулятор