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Resolución de integrales/ 9-x^2/ x^4sqrt(9-x^2)

Integral de x^4sqrt(9-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  3                  
  /                  
 |                   
 |        ________   
 |   4   /      2    
 |  x *\/  9 - x   dx
 |                   
/                    
0
$$\int\limits_{0}^{3} x^{4} \sqrt{9 - x^{2}}\, dx$$ 
Integral(x^4*sqrt(9 - x^2), (x, 0, 3))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=3*sin(_theta), rewritten=729*sin(_theta)**4*cos(_theta)**2, substep=ConstantTimesRule(constant=729, other=sin(_theta)**4*cos(_theta)**2, substep=RewriteRule(rewritten=(1/2 - cos(2*_theta)/2)**2*(cos(2*_theta)/2 + 1/2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=1/16, other=cos(_u)**3, substep=RewriteRule(rewritten=(1 - sin(_u)**2)*cos(_u), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(_u), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1, context=1, symbol=_u), ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2, symbol=_u)], context=1 - _u**2, symbol=_u), context=(1 - sin(_u)**2)*cos(_u), symbol=_u), RewriteRule(rewritten=-sin(_u)**2*cos(_u) + cos(_u), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u), symbol=_u), TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u)], context=-sin(_u)**2*cos(_u) + cos(_u), symbol=_u), context=(1 - sin(_u)**2)*cos(_u), symbol=_u), RewriteRule(rewritten=-sin(_u)**2*cos(_u) + cos(_u), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u), symbol=_u), TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u)], context=-sin(_u)**2*cos(_u) + cos(_u), symbol=_u), context=(1 - sin(_u)**2)*cos(_u), symbol=_u)], context=(1 - sin(_u)**2)*cos(_u), symbol=_u), context=cos(_u)**3, symbol=_u), context=cos(_u)**3/16, symbol=_u), ConstantTimesRule(constant=-1/16, other=cos(_u)**2, substep=RewriteRule(rewritten=cos(2*_u)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(2*_u), substep=URule(u_var=_u, u_func=2*_u, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_u), symbol=_u), context=cos(2*_u)/2, symbol=_u), ConstantRule(constant=1/2, context=1/2, symbol=_u)], context=cos(2*_u)/2 + 1/2, symbol=_u), context=cos(_u)**2, symbol=_u), context=-cos(_u)**2/16, symbol=_u), ConstantTimesRule(constant=-1/16, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=-cos(_u)/16, symbol=_u), ConstantRule(constant=1/16, context=1/16, symbol=_u)], context=cos(_u)**3/16 - cos(_u)**2/16 - cos(_u)/16 + 1/16, symbol=_u), context=(1/2 - cos(2*_theta)/2)**2*(cos(2*_theta)/2 + 1/2), symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**3/8 - cos(2*_theta)**2/8 - cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta)], context=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta)], context=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=-1/8, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=-cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=-1/8, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 - cos(2*_theta)**2/8 - cos(2*_theta)/8 + 1/8, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2*(cos(2*_theta)/2 + 1/2), symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**3/8 - cos(2*_theta)**2/8 - cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta)], context=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta)], context=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=-1/8, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, other=cos(4*_theta), substep=URule(u_var=_u, u_func=4*_theta, constant=1/4, substep=ConstantTimesRule(constant=1/4, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(4*_theta), symbol=_theta), context=cos(4*_theta)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=-cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=-1/8, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=cos(2*_theta)**3/8 - cos(2*_theta)**2/8 - cos(2*_theta)/8 + 1/8, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2*(cos(2*_theta)/2 + 1/2), symbol=_theta)], context=(1/2 - cos(2*_theta)/2)**2*(cos(2*_theta)/2 + 1/2), symbol=_theta), context=sin(_theta)**4*cos(_theta)**2, symbol=_theta), context=729*sin(_theta)**4*cos(_theta)**2, symbol=_theta), restriction=(x > -3) & (x < 3), context=x**4*sqrt(9 - x**2), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
  /                                                                                                            
 |                                                                                                             
 |       ________          //        /x\              3/2          ________                                   \
 |  4   /      2           ||729*asin|-|    3 /     2\            /      2  /       2\                        |
 | x *\/  9 - x   dx = C + |<        \3/   x *\9 - x /      9*x*\/  9 - x  *\9 - 2*x /                        |
 |                         ||----------- - -------------- - --------------------------  for And(x > -3, x < 3)|
/                          \\     16             6                      16                                    /
$$\int x^{4} \sqrt{9 - x^{2}}\, dx = C + \begin{cases} - \frac{x^{3} \left(9 - x^{2}\right)^{\frac{3}{2}}}{6} - \frac{9 x \left(9 - 2 x^{2}\right) \sqrt{9 - x^{2}}}{16} + \frac{729 \operatorname{asin}{\left(\frac{x}{3} \right)}}{16} & \text{for}\: x > -3 \wedge x < 3 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
  3                                                                                                                                                                                
  /                                                                                                                                                                                
 |                                                                                                                                                                                 
 |  /                                                   2                2                4               8                 6                6                 4           2       
 |  |        243*I              729*I            729*I*x           81*I*x           75*I*x             I*x             7*I*x           15*I*x            27*I*x           x        
 |  |- ----------------- + --------------- - --------------- - --------------- - -------------- - -------------- + -------------- + -------------- + ---------------  for -- > 1   
 |  |          _________         _________               3/2         _________        _________              3/2        _________              3/2               3/2      9        
 |  |         /       2         /       2       /      2\           /       2        /       2      /      2\          /       2      /      2\         /      2\                  
 |  |        /       x     16*\/  -9 + x     16*\-9 + x /      16*\/  -9 + x     8*\/  -9 + x     6*\-9 + x /      6*\/  -9 + x     8*\-9 + x /      16*\-9 + x /                  
 |  |  16*  /   -1 + --                                                                                                                                                            
 |  |     \/         9                                                                                                                                                             
 |  <                                                                                                                                                                            dx
 |  |                                                   2               6              8                6               4                4               2                         
 |  |           729               243              729*x             7*x              x             15*x            27*x             75*x            81*x                          
 |  |    - -------------- + ---------------- - -------------- - ------------- - ------------- + ------------- + -------------- + ------------- + --------------       otherwise    
 |  |            ________           ________              3/2        ________             3/2             3/2              3/2        ________         ________                    
 |  |           /      2           /      2       /     2\          /      2      /     2\        /     2\         /     2\          /      2         /      2                     
 |  |      16*\/  9 - x           /      x     16*\9 - x /      6*\/  9 - x     6*\9 - x /      8*\9 - x /      16*\9 - x /      8*\/  9 - x     16*\/  9 - x                      
 |  |                       16*  /   1 - --                                                                                                                                        
 |  \                          \/        9                                                                                                                                         
 |                                                                                                                                                                                 
/                                                                                                                                                                                  
0
$$\int\limits_{0}^{3} \begin{cases} - \frac{i x^{8}}{6 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 \sqrt{x^{2} - 9}} + \frac{15 i x^{6}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{75 i x^{4}}{8 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{16 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{16 \sqrt{x^{2} - 9}} - \frac{729 i x^{2}}{16 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{729 i}{16 \sqrt{x^{2} - 9}} - \frac{243 i}{16 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{8}}{6 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 \sqrt{9 - x^{2}}} + \frac{15 x^{6}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{75 x^{4}}{8 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{16 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{16 \sqrt{9 - x^{2}}} - \frac{729 x^{2}}{16 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{729}{16 \sqrt{9 - x^{2}}} + \frac{243}{16 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$ 
=
= 
  3                                                                                                                                                                                
  /                                                                                                                                                                                
 |                                                                                                                                                                                 
 |  /                                                   2                2                4               8                 6                6                 4           2       
 |  |        243*I              729*I            729*I*x           81*I*x           75*I*x             I*x             7*I*x           15*I*x            27*I*x           x        
 |  |- ----------------- + --------------- - --------------- - --------------- - -------------- - -------------- + -------------- + -------------- + ---------------  for -- > 1   
 |  |          _________         _________               3/2         _________        _________              3/2        _________              3/2               3/2      9        
 |  |         /       2         /       2       /      2\           /       2        /       2      /      2\          /       2      /      2\         /      2\                  
 |  |        /       x     16*\/  -9 + x     16*\-9 + x /      16*\/  -9 + x     8*\/  -9 + x     6*\-9 + x /      6*\/  -9 + x     8*\-9 + x /      16*\-9 + x /                  
 |  |  16*  /   -1 + --                                                                                                                                                            
 |  |     \/         9                                                                                                                                                             
 |  <                                                                                                                                                                            dx
 |  |                                                   2               6              8                6               4                4               2                         
 |  |           729               243              729*x             7*x              x             15*x            27*x             75*x            81*x                          
 |  |    - -------------- + ---------------- - -------------- - ------------- - ------------- + ------------- + -------------- + ------------- + --------------       otherwise    
 |  |            ________           ________              3/2        ________             3/2             3/2              3/2        ________         ________                    
 |  |           /      2           /      2       /     2\          /      2      /     2\        /     2\         /     2\          /      2         /      2                     
 |  |      16*\/  9 - x           /      x     16*\9 - x /      6*\/  9 - x     6*\9 - x /      8*\9 - x /      16*\9 - x /      8*\/  9 - x     16*\/  9 - x                      
 |  |                       16*  /   1 - --                                                                                                                                        
 |  \                          \/        9                                                                                                                                         
 |                                                                                                                                                                                 
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0
$$\int\limits_{0}^{3} \begin{cases} - \frac{i x^{8}}{6 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{7 i x^{6}}{6 \sqrt{x^{2} - 9}} + \frac{15 i x^{6}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{75 i x^{4}}{8 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{16 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{16 \sqrt{x^{2} - 9}} - \frac{729 i x^{2}}{16 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{729 i}{16 \sqrt{x^{2} - 9}} - \frac{243 i}{16 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{8}}{6 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{7 x^{6}}{6 \sqrt{9 - x^{2}}} + \frac{15 x^{6}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{75 x^{4}}{8 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{16 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{16 \sqrt{9 - x^{2}}} - \frac{729 x^{2}}{16 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{729}{16 \sqrt{9 - x^{2}}} + \frac{243}{16 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$ 
Integral(Piecewise((-243*i/(16*sqrt(-1 + x^2/9)) + 729*i/(16*sqrt(-9 + x^2)) - 729*i*x^2/(16*(-9 + x^2)^(3/2)) - 81*i*x^2/(16*sqrt(-9 + x^2)) - 75*i*x^4/(8*sqrt(-9 + x^2)) - i*x^8/(6*(-9 + x^2)^(3/2)) + 7*i*x^6/(6*sqrt(-9 + x^2)) + 15*i*x^6/(8*(-9 + x^2)^(3/2)) + 27*i*x^4/(16*(-9 + x^2)^(3/2)), x^2/9 > 1), (-729/(16*sqrt(9 - x^2)) + 243/(16*sqrt(1 - x^2/9)) - 729*x^2/(16*(9 - x^2)^(3/2)) - 7*x^6/(6*sqrt(9 - x^2)) - x^8/(6*(9 - x^2)^(3/2)) + 15*x^6/(8*(9 - x^2)^(3/2)) + 27*x^4/(16*(9 - x^2)^(3/2)) + 75*x^4/(8*sqrt(9 - x^2)) + 81*x^2/(16*sqrt(9 - x^2)), True)), (x, 0, 3))
Respuesta numérica [src] 
71.5694076395925
71.5694076395925

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

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