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Resolución de integrales/ sqrt(1-x*x)/ x^4*(sqrt(1-x*x))

Integral de x^4*(sqrt(1-x*x)) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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 |  x *\/ 1 - x*x  dx
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∫01x4−xx+1 dx\int\limits_{0}^{1} x^{4} \sqrt{- x x + 1}\, dx 
Integral(x^4*sqrt(1 - x*x), (x, 0, 1))
Respuesta (Indefinida) [src] 
  /                                                                                                      
 |                         //                     3/2        ________                                   \
 |  4   _________          ||           3 /     2\          /      2  /       2\                        |
 | x *\/ 1 - x*x  dx = C + | -1, x < 1)|
/                          \\   16           6                     16                                   /
∫x4−xx+1 dx=C+{−x3(1−x2)326−x(1−2x2)1−x216+asin⁡(x)16for x>−1∧x<1\int x^{4} \sqrt{- x x + 1}\, dx = C + \begin{cases} - \frac{x^{3} \left(1 - x^{2}\right)^{\frac{3}{2}}}{6} - \frac{x \left(1 - 2 x^{2}\right) \sqrt{1 - x^{2}}}{16} + \frac{\operatorname{asin}{\left(x \right)}}{16} & \text{for}\: x > -1 \wedge x < 1 \end{cases}
Simplificar
Gráfica
0.001.000.100.200.300.400.500.600.700.800.900.00.5
Trazar el gráfico f(x)
Respuesta [src] 
  1                                                                                                                                            
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 |  /            4               8                 2                 2                 4                  6                6                   
 |  |      25*I*x             I*x               I*x               I*x               I*x              5*I*x            7*I*x            2       
 |  |- --------------- - -------------- - --------------- - --------------- + --------------- + --------------- + --------------  for x  > 1   
 |  |        _________              3/2               3/2         _________               3/2               3/2        _________               
 |  |       /       2      /      2\         /      2\           /       2       /      2\         /      2\          /       2                
 |  |  24*\/  -1 + x     6*\-1 + x /      16*\-1 + x /      16*\/  -1 + x     48*\-1 + x /      24*\-1 + x /      6*\/  -1 + x                 
 |  <                                                                                                                                        dx
 |  |             6              8               2                2                4                 6                4                        
 |  |          7*x              x               x                x                x               5*x             25*x                         
 |  |   - ------------- - ------------- - -------------- + -------------- + -------------- + -------------- + --------------      otherwise    
 |  |          ________             3/2              3/2         ________              3/2              3/2         ________                   
 |  |         /      2      /     2\         /     2\           /      2       /     2\         /     2\           /      2                    
 |  \     6*\/  1 - x     6*\1 - x /      16*\1 - x /      16*\/  1 - x     48*\1 - x /      24*\1 - x /      24*\/  1 - x                     
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∫01{−ix86(x2−1)32+7ix66x2−1+5ix624(x2−1)32−25ix424x2−1+ix448(x2−1)32−ix216x2−1−ix216(x2−1)32for x2>1−x86(1−x2)32−7x661−x2+5x624(1−x2)32+25x4241−x2+x448(1−x2)32+x2161−x2−x216(1−x2)32otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i x^{8}}{6 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 \sqrt{x^{2} - 1}} + \frac{5 i x^{6}}{24 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{25 i x^{4}}{24 \sqrt{x^{2} - 1}} + \frac{i x^{4}}{48 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{i x^{2}}{16 \sqrt{x^{2} - 1}} - \frac{i x^{2}}{16 \left(x^{2} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{2} > 1 \\- \frac{x^{8}}{6 \left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 \sqrt{1 - x^{2}}} + \frac{5 x^{6}}{24 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{25 x^{4}}{24 \sqrt{1 - x^{2}}} + \frac{x^{4}}{48 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{x^{2}}{16 \sqrt{1 - x^{2}}} - \frac{x^{2}}{16 \left(1 - x^{2}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx 
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  1                                                                                                                                            
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 |  /            4               8                 2                 2                 4                  6                6                   
 |  |      25*I*x             I*x               I*x               I*x               I*x              5*I*x            7*I*x            2       
 |  |- --------------- - -------------- - --------------- - --------------- + --------------- + --------------- + --------------  for x  > 1   
 |  |        _________              3/2               3/2         _________               3/2               3/2        _________               
 |  |       /       2      /      2\         /      2\           /       2       /      2\         /      2\          /       2                
 |  |  24*\/  -1 + x     6*\-1 + x /      16*\-1 + x /      16*\/  -1 + x     48*\-1 + x /      24*\-1 + x /      6*\/  -1 + x                 
 |  <                                                                                                                                        dx
 |  |             6              8               2                2                4                 6                4                        
 |  |          7*x              x               x                x                x               5*x             25*x                         
 |  |   - ------------- - ------------- - -------------- + -------------- + -------------- + -------------- + --------------      otherwise    
 |  |          ________             3/2              3/2         ________              3/2              3/2         ________                   
 |  |         /      2      /     2\         /     2\           /      2       /     2\         /     2\           /      2                    
 |  \     6*\/  1 - x     6*\1 - x /      16*\1 - x /      16*\/  1 - x     48*\1 - x /      24*\1 - x /      24*\/  1 - x                     
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0
∫01{−ix86(x2−1)32+7ix66x2−1+5ix624(x2−1)32−25ix424x2−1+ix448(x2−1)32−ix216x2−1−ix216(x2−1)32for x2>1−x86(1−x2)32−7x661−x2+5x624(1−x2)32+25x4241−x2+x448(1−x2)32+x2161−x2−x216(1−x2)32otherwise dx\int\limits_{0}^{1} \begin{cases} - \frac{i x^{8}}{6 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 \sqrt{x^{2} - 1}} + \frac{5 i x^{6}}{24 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{25 i x^{4}}{24 \sqrt{x^{2} - 1}} + \frac{i x^{4}}{48 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{i x^{2}}{16 \sqrt{x^{2} - 1}} - \frac{i x^{2}}{16 \left(x^{2} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{2} > 1 \\- \frac{x^{8}}{6 \left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 \sqrt{1 - x^{2}}} + \frac{5 x^{6}}{24 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{25 x^{4}}{24 \sqrt{1 - x^{2}}} + \frac{x^{4}}{48 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{x^{2}}{16 \sqrt{1 - x^{2}}} - \frac{x^{2}}{16 \left(1 - x^{2}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx 
Integral(Piecewise((-25*i*x^4/(24*sqrt(-1 + x^2)) - i*x^8/(6*(-1 + x^2)^(3/2)) - i*x^2/(16*(-1 + x^2)^(3/2)) - i*x^2/(16*sqrt(-1 + x^2)) + i*x^4/(48*(-1 + x^2)^(3/2)) + 5*i*x^6/(24*(-1 + x^2)^(3/2)) + 7*i*x^6/(6*sqrt(-1 + x^2)), x^2 > 1), (-7*x^6/(6*sqrt(1 - x^2)) - x^8/(6*(1 - x^2)^(3/2)) - x^2/(16*(1 - x^2)^(3/2)) + x^2/(16*sqrt(1 - x^2)) + x^4/(48*(1 - x^2)^(3/2)) + 5*x^6/(24*(1 - x^2)^(3/2)) + 25*x^4/(24*sqrt(1 - x^2)), True)), (x, 0, 1))
Respuesta numérica [src] 
0.098174770424681
0.098174770424681

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

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