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

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

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

  1. Ahora simplificar:

  2. Añadimos la constante de integración:


Respuesta:

Respuesta (Indefinida) [src]
  /                                                                                            
 |                                                                                             
 |         ________          //         /x\        ________                                   \
 |        /      2           ||  81*asin|-|       /      2  /       2\                        |
 | x*-x*\/  9 - x   dx = C + |<         \3/   x*\/  9 - x  *\9 - 2*x /                        |
 |                           ||- ---------- + ------------------------  for And(x > -3, x < 3)|
/                            \\      8                   8                                    /
$$\int x - x \sqrt{9 - x^{2}}\, dx = C + \begin{cases} \frac{x \left(9 - 2 x^{2}\right) \sqrt{9 - x^{2}}}{8} - \frac{81 \operatorname{asin}{\left(\frac{x}{3} \right)}}{8} & \text{for}\: x > -3 \wedge x < 3 \end{cases}$$
Gráfica
Respuesta [src]
  3                                                                                                                                         
  /                                                                                                                                         
 |                                                                                                                                          
 |  /                                               4                4               6                 2                2           2       
 |  |       81*I              27*I            27*I*x            5*I*x             I*x            81*I*x           81*I*x           x        
 |  |- -------------- + ---------------- - -------------- - -------------- + -------------- + -------------- + --------------  for -- > 1   
 |  |       _________          _________              3/2        _________              3/2              3/2        _________      9        
 |  |      /       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                                                                                                      
 |  <                                                                                                                                     dx
 |  |                                               2               4              6                4               2                       
 |  |            27               81            81*x            27*x              x              5*x            81*x                        
 |  |   - --------------- + ------------- - ------------- - ------------- + ------------- + ------------- + -------------      otherwise    
 |  |            ________        ________        ________             3/2             3/2        ________             3/2                   
 |  |           /      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                                                                                                                     
 |                                                                                                                                          
/                                                                                                                                           
0                                                                                                                                           
$$\int\limits_{0}^{3} \begin{cases} \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} - \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} + \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i}{8 \sqrt{x^{2} - 9}} + \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\\frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} - \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} + \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81}{8 \sqrt{9 - x^{2}}} - \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
=
=
  3                                                                                                                                         
  /                                                                                                                                         
 |                                                                                                                                          
 |  /                                               4                4               6                 2                2           2       
 |  |       81*I              27*I            27*I*x            5*I*x             I*x            81*I*x           81*I*x           x        
 |  |- -------------- + ---------------- - -------------- - -------------- + -------------- + -------------- + --------------  for -- > 1   
 |  |       _________          _________              3/2        _________              3/2              3/2        _________      9        
 |  |      /       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                                                                                                      
 |  <                                                                                                                                     dx
 |  |                                               2               4              6                4               2                       
 |  |            27               81            81*x            27*x              x              5*x            81*x                        
 |  |   - --------------- + ------------- - ------------- - ------------- + ------------- + ------------- + -------------      otherwise    
 |  |            ________        ________        ________             3/2             3/2        ________             3/2                   
 |  |           /      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                                                                                                                     
 |                                                                                                                                          
/                                                                                                                                           
0                                                                                                                                           
$$\int\limits_{0}^{3} \begin{cases} \frac{i x^{6}}{4 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{5 i x^{4}}{4 \sqrt{x^{2} - 9}} - \frac{27 i x^{4}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} + \frac{81 i x^{2}}{8 \sqrt{x^{2} - 9}} + \frac{81 i x^{2}}{8 \left(x^{2} - 9\right)^{\frac{3}{2}}} - \frac{81 i}{8 \sqrt{x^{2} - 9}} + \frac{27 i}{8 \sqrt{\frac{x^{2}}{9} - 1}} & \text{for}\: \frac{x^{2}}{9} > 1 \\\frac{x^{6}}{4 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{5 x^{4}}{4 \sqrt{9 - x^{2}}} - \frac{27 x^{4}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} - \frac{81 x^{2}}{8 \sqrt{9 - x^{2}}} + \frac{81 x^{2}}{8 \left(9 - x^{2}\right)^{\frac{3}{2}}} + \frac{81}{8 \sqrt{9 - x^{2}}} - \frac{27}{8 \sqrt{1 - \frac{x^{2}}{9}}} & \text{otherwise} \end{cases}\, dx$$
Integral(Piecewise((-81*i/(8*sqrt(-9 + x^2)) + 27*i/(8*sqrt(-1 + x^2/9)) - 27*i*x^4/(8*(-9 + x^2)^(3/2)) - 5*i*x^4/(4*sqrt(-9 + x^2)) + i*x^6/(4*(-9 + x^2)^(3/2)) + 81*i*x^2/(8*(-9 + x^2)^(3/2)) + 81*i*x^2/(8*sqrt(-9 + x^2)), x^2/9 > 1), (-27/(8*sqrt(1 - x^2/9)) + 81/(8*sqrt(9 - x^2)) - 81*x^2/(8*sqrt(9 - x^2)) - 27*x^4/(8*(9 - x^2)^(3/2)) + x^6/(4*(9 - x^2)^(3/2)) + 5*x^4/(4*sqrt(9 - x^2)) + 81*x^2/(8*(9 - x^2)^(3/2)), True)), (x, 0, 3))
Respuesta numérica [src]
-15.9043128087983
-15.9043128087983

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