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Integral de y=x^2*sqrt(4-x^2) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

  1. Ahora simplificar:

  2. Añadimos la constante de integración:


Respuesta:

Respuesta (Indefinida) [src]
  /                                                                                     
 |                         //                 ________ /     2\                        \
 |       ________          ||                /      2  |    x |                        |
 |  2   /      2           ||            x*\/  4 - x  *|1 - --|                        |
 | x *\/  4 - x   dx = C + |<      /x\                 \    2 /                        |
 |                         ||2*asin|-| - ----------------------  for And(x > -2, x < 2)|
/                          ||      \2/             2                                   |
                           \\                                                          /
$$\int x^{2} \sqrt{4 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(1 - \frac{x^{2}}{2}\right) \sqrt{4 - x^{2}}}{2} + 2 \operatorname{asin}{\left(\frac{x}{2} \right)} & \text{for}\: x > -2 \wedge x < 2 \end{cases}$$
Gráfica
Respuesta [src]
  2                                                                                                                                   
  /                                                                                                                                   
 |                                                                                                                                    
 |  /                                          2               2               6                 4                4           2       
 |  |        I              2*I           2*I*x           9*I*x             I*x             3*I*x            5*I*x           x        
 |  |- -------------- + ------------ - ------------ - -------------- - -------------- + -------------- + --------------  for -- > 1   
 |  |       _________      _________            3/2        _________              3/2              3/2        _________      4        
 |  |      /       2      /       2    /      2\          /       2      /      2\        /      2\          /       2                
 |  |     /       x     \/  -4 + x     \-4 + x /      2*\/  -4 + x     4*\-4 + x /      2*\-4 + x /      4*\/  -4 + x                 
 |  |    /   -1 + --                                                                                                                  
 |  |  \/         4                                                                                                                   
 |  <                                                                                                                               dx
 |  |                                         2              4              6                4               2                        
 |  |          1              2            2*x            5*x              x              3*x             9*x                         
 |  |    ------------- - ----------- - ----------- - ------------- - ------------- + ------------- + -------------       otherwise    
 |  |         ________      ________           3/2        ________             3/2             3/2        ________                    
 |  |        /      2      /      2    /     2\          /      2      /     2\        /     2\          /      2                     
 |  |       /      x     \/  4 - x     \4 - x /      4*\/  4 - x     4*\4 - x /      2*\4 - x /      2*\/  4 - x                      
 |  |      /   1 - --                                                                                                                 
 |  \    \/        4                                                                                                                  
 |                                                                                                                                    
/                                                                                                                                     
0                                                                                                                                     
$$\int\limits_{0}^{2} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 4}} + \frac{3 i x^{4}}{2 \left(x^{2} - 4\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{2 \sqrt{x^{2} - 4}} - \frac{2 i x^{2}}{\left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{2 i}{\sqrt{x^{2} - 4}} - \frac{i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\- \frac{x^{6}}{4 \left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{4 - x^{2}}} + \frac{3 x^{4}}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{2 \sqrt{4 - x^{2}}} - \frac{2 x^{2}}{\left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{4 - x^{2}}} + \frac{1}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$
=
=
  2                                                                                                                                   
  /                                                                                                                                   
 |                                                                                                                                    
 |  /                                          2               2               6                 4                4           2       
 |  |        I              2*I           2*I*x           9*I*x             I*x             3*I*x            5*I*x           x        
 |  |- -------------- + ------------ - ------------ - -------------- - -------------- + -------------- + --------------  for -- > 1   
 |  |       _________      _________            3/2        _________              3/2              3/2        _________      4        
 |  |      /       2      /       2    /      2\          /       2      /      2\        /      2\          /       2                
 |  |     /       x     \/  -4 + x     \-4 + x /      2*\/  -4 + x     4*\-4 + x /      2*\-4 + x /      4*\/  -4 + x                 
 |  |    /   -1 + --                                                                                                                  
 |  |  \/         4                                                                                                                   
 |  <                                                                                                                               dx
 |  |                                         2              4              6                4               2                        
 |  |          1              2            2*x            5*x              x              3*x             9*x                         
 |  |    ------------- - ----------- - ----------- - ------------- - ------------- + ------------- + -------------       otherwise    
 |  |         ________      ________           3/2        ________             3/2             3/2        ________                    
 |  |        /      2      /      2    /     2\          /      2      /     2\        /     2\          /      2                     
 |  |       /      x     \/  4 - x     \4 - x /      4*\/  4 - x     4*\4 - x /      2*\4 - x /      2*\/  4 - x                      
 |  |      /   1 - --                                                                                                                 
 |  \    \/        4                                                                                                                  
 |                                                                                                                                    
/                                                                                                                                     
0                                                                                                                                     
$$\int\limits_{0}^{2} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 4}} + \frac{3 i x^{4}}{2 \left(x^{2} - 4\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{2 \sqrt{x^{2} - 4}} - \frac{2 i x^{2}}{\left(x^{2} - 4\right)^{\frac{3}{2}}} + \frac{2 i}{\sqrt{x^{2} - 4}} - \frac{i}{\sqrt{\frac{x^{2}}{4} - 1}} & \text{for}\: \frac{x^{2}}{4} > 1 \\- \frac{x^{6}}{4 \left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{4 - x^{2}}} + \frac{3 x^{4}}{2 \left(4 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{2 \sqrt{4 - x^{2}}} - \frac{2 x^{2}}{\left(4 - x^{2}\right)^{\frac{3}{2}}} - \frac{2}{\sqrt{4 - x^{2}}} + \frac{1}{\sqrt{1 - \frac{x^{2}}{4}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-i/sqrt(-1 + x^2/4) + 2*i/sqrt(-4 + x^2) - 2*i*x^2/(-4 + x^2)^(3/2) - 9*i*x^2/(2*sqrt(-4 + x^2)) - i*x^6/(4*(-4 + x^2)^(3/2)) + 3*i*x^4/(2*(-4 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-4 + x^2)), x^2/4 > 1), (1/sqrt(1 - x^2/4) - 2/sqrt(4 - x^2) - 2*x^2/(4 - x^2)^(3/2) - 5*x^4/(4*sqrt(4 - x^2)) - x^6/(4*(4 - x^2)^(3/2)) + 3*x^4/(2*(4 - x^2)^(3/2)) + 9*x^2/(2*sqrt(4 - x^2)), True)), (x, 0, 2))
Respuesta numérica [src]
3.14159265358979
3.14159265358979

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