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Resolución de integrales/ 1-x^2/ sqrt(1-x^2)x^2

Integral de sqrt(1-x^2)x^2 dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

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

    TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=1/8 - cos(4*_theta)/8, substep=AddRule(substeps=[ConstantRule(constant=1/8, context=1/8, symbol=_theta), ConstantTimesRule(constant=-1/8, 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=-cos(4*_theta)/8, symbol=_theta)], context=1/8 - cos(4*_theta)/8, symbol=_theta), restriction=(x > -1) & (x < 1), context=x**2*sqrt(1 - x**2), symbol=x)

  1. Ahora simplificar:

  2. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
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 | \/  1 - x  *x  dx = C + | -1, x < 1)|
/                          \\   8                 8                                    /
$$\int x^{2} \sqrt{1 - x^{2}}\, dx = C + \begin{cases} - \frac{x \left(1 - 2 x^{2}\right) \sqrt{1 - x^{2}}}{8} + \frac{\operatorname{asin}{\left(x \right)}}{8} & \text{for}\: x > -1 \wedge x < 1 \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
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 |  /           2               6                2                 4                4                   
 |  |      9*I*x             I*x              I*x             3*I*x            5*I*x            2       
 |  |- -------------- - -------------- - -------------- + -------------- + --------------  for x  > 1   
 |  |       _________              3/2              3/2              3/2        _________               
 |  |      /       2      /      2\        /      2\        /      2\          /       2                
 |  |  8*\/  -1 + x     4*\-1 + x /      8*\-1 + x /      8*\-1 + x /      4*\/  -1 + x                 
 |  <                                                                                                 dx
 |  |            4              6               2                4               2                      
 |  |         5*x              x               x              3*x             9*x                       
 |  |  - ------------- - ------------- - ------------- + ------------- + -------------     otherwise    
 |  |         ________             3/2             3/2             3/2        ________                  
 |  |        /      2      /     2\        /     2\        /     2\          /      2                   
 |  \    4*\/  1 - x     4*\1 - x /      8*\1 - x /      8*\1 - x /      8*\/  1 - x                    
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0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 1}} + \frac{3 i x^{4}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{8 \sqrt{x^{2} - 1}} - \frac{i x^{2}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{2} > 1 \\- \frac{x^{6}}{4 \left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{1 - x^{2}}} + \frac{3 x^{4}}{8 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{8 \sqrt{1 - x^{2}}} - \frac{x^{2}}{8 \left(1 - x^{2}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$ 
=
= 
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 |  /           2               6                2                 4                4                   
 |  |      9*I*x             I*x              I*x             3*I*x            5*I*x            2       
 |  |- -------------- - -------------- - -------------- + -------------- + --------------  for x  > 1   
 |  |       _________              3/2              3/2              3/2        _________               
 |  |      /       2      /      2\        /      2\        /      2\          /       2                
 |  |  8*\/  -1 + x     4*\-1 + x /      8*\-1 + x /      8*\-1 + x /      4*\/  -1 + x                 
 |  <                                                                                                 dx
 |  |            4              6               2                4               2                      
 |  |         5*x              x               x              3*x             9*x                       
 |  |  - ------------- - ------------- - ------------- + ------------- + -------------     otherwise    
 |  |         ________             3/2             3/2             3/2        ________                  
 |  |        /      2      /     2\        /     2\        /     2\          /      2                   
 |  \    4*\/  1 - x     4*\1 - x /      8*\1 - x /      8*\1 - x /      8*\/  1 - x                    
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0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 1}} + \frac{3 i x^{4}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{8 \sqrt{x^{2} - 1}} - \frac{i x^{2}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} & \text{for}\: x^{2} > 1 \\- \frac{x^{6}}{4 \left(1 - x^{2}\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{1 - x^{2}}} + \frac{3 x^{4}}{8 \left(1 - x^{2}\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{8 \sqrt{1 - x^{2}}} - \frac{x^{2}}{8 \left(1 - x^{2}\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$ 
Integral(Piecewise((-9*i*x^2/(8*sqrt(-1 + x^2)) - i*x^6/(4*(-1 + x^2)^(3/2)) - i*x^2/(8*(-1 + x^2)^(3/2)) + 3*i*x^4/(8*(-1 + x^2)^(3/2)) + 5*i*x^4/(4*sqrt(-1 + x^2)), x^2 > 1), (-5*x^4/(4*sqrt(1 - x^2)) - x^6/(4*(1 - x^2)^(3/2)) - x^2/(8*(1 - x^2)^(3/2)) + 3*x^4/(8*(1 - x^2)^(3/2)) + 9*x^2/(8*sqrt(1 - x^2)), True)), (x, 0, 1))
Respuesta numérica [src] 
0.196349540849362
0.196349540849362

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

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