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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src]
  1                  
  /                  
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 |        ________   
 |   2   /      2    
 |  x *\/  9 - x   dx
 |                   
/                    
0                    
$$\int\limits_{0}^{1} x^{2} \sqrt{9 - x^{2}}\, dx$$
Integral(x^2*sqrt(9 - x^2), (x, 0, 1))
Solución detallada

    TrigSubstitutionRule(theta=_theta, func=3*sin(_theta), rewritten=81/8 - 81*cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=81/8, context=81/8, symbol=_theta), ConstantTimesRule(constant=-81/8, 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=-81*cos(4*_theta)/8, symbol=_theta)], context=81/8 - 81*cos(4*_theta)/8, symbol=_theta), restriction=(x > -3) & (x < 3), context=x**2*sqrt(9 - x**2), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:


Respuesta:

Respuesta (Indefinida) [src]
  /                                                                                        
 |                                                                                         
 |       ________          //       /x\        ________                                   \
 |  2   /      2           ||81*asin|-|       /      2  /       2\                        |
 | x *\/  9 - x   dx = C + |<       \3/   x*\/  9 - x  *\9 - 2*x /                        |
 |                         ||---------- - ------------------------  for And(x > -3, x < 3)|
/                          \\    8                   8                                    /
$$\int x^{2} \sqrt{9 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(9 - 2 x^{2}\right) \sqrt{9 - x^{2}}}{8} + \frac{81 \operatorname{asin}{\left(\frac{x}{3} \right)}}{8} & \text{for}\: x > -3 \wedge x < 3 \end{cases}$$
Gráfica
Respuesta [src]
  1                                                                                                                                         
  /                                                                                                                                         
 |                                                                                                                                          
 |  /                                               2                2               6                 4                4           2       
 |  |        27*I              81*I           81*I*x           81*I*x             I*x             5*I*x           27*I*x           x        
 |  |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + --------------  for -- > 1   
 |  |         _________        _________              3/2        _________              3/2        _________              3/2      9        
 |  |        /       2        /       2      /      2\          /       2      /      2\          /       2      /      2\                  
 |  |       /       x     8*\/  -9 + x     8*\-9 + x /      8*\/  -9 + x     4*\-9 + x /      4*\/  -9 + x     8*\-9 + x /                  
 |  |  8*  /   -1 + --                                                                                                                      
 |  |    \/         9                                                                                                                       
 |  <                                                                                                                                     dx
 |  |                                               2               4              6                4               2                       
 |  |           81               27             81*x             5*x              x             27*x            81*x                        
 |  |   - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + -------------      otherwise    
 |  |          ________          ________             3/2        ________             3/2             3/2        ________                   
 |  |         /      2          /      2      /     2\          /      2      /     2\        /     2\          /      2                    
 |  |     8*\/  9 - x          /      x     8*\9 - x /      4*\/  9 - x     4*\9 - x /      8*\9 - x /      8*\/  9 - x                     
 |  |                     8*  /   1 - --                                                                                                    
 |  \                       \/        9                                                                                                     
 |                                                                                                                                          
/                                                                                                                                           
0                                                                                                                                           
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i}{8 \sqrt{x^{2} - 9}} - \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81}{8 \sqrt{9 - x^{2}}} + \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
=
=
  1                                                                                                                                         
  /                                                                                                                                         
 |                                                                                                                                          
 |  /                                               2                2               6                 4                4           2       
 |  |        27*I              81*I           81*I*x           81*I*x             I*x             5*I*x           27*I*x           x        
 |  |- ---------------- + -------------- - -------------- - -------------- - -------------- + -------------- + --------------  for -- > 1   
 |  |         _________        _________              3/2        _________              3/2        _________              3/2      9        
 |  |        /       2        /       2      /      2\          /       2      /      2\          /       2      /      2\                  
 |  |       /       x     8*\/  -9 + x     8*\-9 + x /      8*\/  -9 + x     4*\-9 + x /      4*\/  -9 + x     8*\-9 + x /                  
 |  |  8*  /   -1 + --                                                                                                                      
 |  |    \/         9                                                                                                                       
 |  <                                                                                                                                     dx
 |  |                                               2               4              6                4               2                       
 |  |           81               27             81*x             5*x              x             27*x            81*x                        
 |  |   - ------------- + --------------- - ------------- - ------------- - ------------- + ------------- + -------------      otherwise    
 |  |          ________          ________             3/2        ________             3/2             3/2        ________                   
 |  |         /      2          /      2      /     2\          /      2      /     2\        /     2\          /      2                    
 |  |     8*\/  9 - x          /      x     8*\9 - x /      4*\/  9 - x     4*\9 - x /      8*\9 - x /      8*\/  9 - x                     
 |  |                     8*  /   1 - --                                                                                                    
 |  \                       \/        9                                                                                                     
 |                                                                                                                                          
/                                                                                                                                           
0                                                                                                                                           
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} + \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} - \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i}{8 \sqrt{x^{2} - 9}} - \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\- \frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} + \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} - \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81}{8 \sqrt{9 - x^{2}}} + \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-27*i/(8*sqrt(-1 + x^2/9)) + 81*i/(8*sqrt(-9 + x^2)) - 81*i*x^2/(8*(-9 + x^2)^(3/2)) - 81*i*x^2/(8*sqrt(-9 + x^2)) - i*x^6/(4*(-9 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-9 + x^2)) + 27*i*x^4/(8*(-9 + x^2)^(3/2)), x^2/9 > 1), (-81/(8*sqrt(9 - x^2)) + 27/(8*sqrt(1 - x^2/9)) - 81*x^2/(8*(9 - x^2)^(3/2)) - 5*x^4/(4*sqrt(9 - x^2)) - x^6/(4*(9 - x^2)^(3/2)) + 27*x^4/(8*(9 - x^2)^(3/2)) + 81*x^2/(8*sqrt(9 - x^2)), True)), (x, 0, 1))
Respuesta numérica [src]
0.965974974070068
0.965974974070068

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