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Resolución de integrales/ x^2+9/ 6*x^2/ 1/(x^4-6*x^2+9)

Integral de 1/(x^4-6*x^2+9) dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
  1                 
  /                 
 |                  
 |        1         
 |  ------------- dx
 |   4      2       
 |  x  - 6*x  + 9   
 |                  
/                   
0
$$\int\limits_{0}^{1} \frac{1}{\left(x^{4} - 6 x^{2}\right) + 9}\, dx$$ 
Integral(1/(x^4 - 6*x^2 + 9), (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    TrigSubstitutionRule(theta=_theta, func=sqrt(3)*sec(_theta), rewritten=sqrt(3)*sec(_theta)/(9*tan(_theta)**3), substep=ConstantTimesRule(constant=sqrt(3)/9, other=sec(_theta)/tan(_theta)**3, substep=RewriteRule(rewritten=tan(_theta)*sec(_theta)/(sec(_theta)**2 - 1)**2, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=RewriteRule(rewritten=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=1/(_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u + 1), symbol=_u), context=1/(4*(_u + 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u + 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u + 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u)], context=(_u + 1)**(-2), symbol=_u), context=1/(4*(_u + 1)**2), symbol=_u), ConstantTimesRule(constant=-1/4, other=1/(_u - 1), substep=URule(u_var=_u, u_func=_u - 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u - 1), symbol=_u), context=-1/(4*(_u - 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u - 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u - 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u)], context=(_u - 1)**(-2), symbol=_u), context=1/(4*(_u - 1)**2), symbol=_u)], context=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), symbol=_u), context=1/(_u**4 - 2*_u**2 + 1), symbol=_u), context=tan(_theta)*sec(_theta)/(sec(_theta)**2 - 1)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)/(sec(_theta)**4 - 2*sec(_theta)**2 + 1), substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=RewriteRule(rewritten=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=1/(_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u + 1), symbol=_u), context=1/(4*(_u + 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u + 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u + 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u)], context=(_u + 1)**(-2), symbol=_u), context=1/(4*(_u + 1)**2), symbol=_u), ConstantTimesRule(constant=-1/4, other=1/(_u - 1), substep=URule(u_var=_u, u_func=_u - 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u - 1), symbol=_u), context=-1/(4*(_u - 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u - 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u - 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u)], context=(_u - 1)**(-2), symbol=_u), context=1/(4*(_u - 1)**2), symbol=_u)], context=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), symbol=_u), context=1/(_u**4 - 2*_u**2 + 1), symbol=_u), context=tan(_theta)*sec(_theta)/(sec(_theta)**4 - 2*sec(_theta)**2 + 1), symbol=_theta), context=tan(_theta)*sec(_theta)/(sec(_theta)**2 - 1)**2, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)/(sec(_theta)**4 - 2*sec(_theta)**2 + 1), substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=RewriteRule(rewritten=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=1/(_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u + 1), symbol=_u), context=1/(4*(_u + 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u + 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u + 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 + 2*_u + 1), substep=URule(u_var=_u, u_func=_u + 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=1/(2*_u + (_u - 1)**2 - 1), symbol=_u), context=(_u + 1)**(-2), symbol=_u)], context=(_u + 1)**(-2), symbol=_u), context=1/(4*(_u + 1)**2), symbol=_u), ConstantTimesRule(constant=-1/4, other=1/(_u - 1), substep=URule(u_var=_u, u_func=_u - 1, constant=1, substep=ReciprocalRule(func=_u, context=1/_u, symbol=_u), context=1/(_u - 1), symbol=_u), context=-1/(4*(_u - 1)), symbol=_u), ConstantTimesRule(constant=1/4, other=(_u - 1)**(-2), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=_u - 1, constant=1, substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u), RewriteRule(rewritten=1/(_u**2 - 2*_u + 1), substep=URule(u_var=_u, u_func=_u - 1, constant=None, substep=RewriteRule(rewritten=_u**(-2), substep=PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=1/(-2*_u + (_u + 1)**2 - 1), symbol=_u), context=(_u - 1)**(-2), symbol=_u)], context=(_u - 1)**(-2), symbol=_u), context=1/(4*(_u - 1)**2), symbol=_u)], context=1/(4*(_u + 1)) + 1/(4*(_u + 1)**2) - 1/(4*(_u - 1)) + 1/(4*(_u - 1)**2), symbol=_u), context=1/(_u**4 - 2*_u**2 + 1), symbol=_u), context=tan(_theta)*sec(_theta)/(sec(_theta)**4 - 2*sec(_theta)**2 + 1), symbol=_theta), context=tan(_theta)*sec(_theta)/(sec(_theta)**2 - 1)**2, symbol=_theta)], context=tan(_theta)*sec(_theta)/(sec(_theta)**2 - 1)**2, symbol=_theta), context=sec(_theta)/tan(_theta)**3, symbol=_theta), context=sqrt(3)*sec(_theta)/(9*tan(_theta)**3), symbol=_theta), restriction=(x < sqrt(3)) & (x > -sqrt(3)), context=(x**2 - 3)**(-2), symbol=x)

  2. Ahora simplificar:

  3. Añadimos la constante de integración:

Respuesta:

Respuesta (Indefinida) [src] 
                          //      /                                          /         ___\      /        ___\\                                \
                          ||      |                                          |     x*\/ 3 |      |    x*\/ 3 ||                                |
  /                       ||      |                                       log|-1 + -------|   log|1 + -------||                                |
 |                        ||  ___ |         1                 1              \        3   /      \       3   /|                                |
 |       1                ||\/ 3 *|- --------------- - ---------------- - ----------------- + ----------------|                                |
 | ------------- dx = C + |<      |    /        ___\     /         ___\           4                  4        |                                |
 |  4      2              ||      |    |    x*\/ 3 |     |     x*\/ 3 |                                       |                                |
 | x  - 6*x  + 9          ||      |  4*|1 + -------|   4*|-1 + -------|                                       |                                |
 |                        ||      \    \       3   /     \        3   /                                       /         /       ___        ___\|
/                         ||-----------------------------------------------------------------------------------  for And\x > -\/ 3 , x < \/ 3 /|
                          \\                                         9                                                                         /
$$\int \frac{1}{\left(x^{4} - 6 x^{2}\right) + 9}\, dx = C + \begin{cases} \frac{\sqrt{3} \left(- \frac{\log{\left(\frac{\sqrt{3} x}{3} - 1 \right)}}{4} + \frac{\log{\left(\frac{\sqrt{3} x}{3} + 1 \right)}}{4} - \frac{1}{4 \left(\frac{\sqrt{3} x}{3} + 1\right)} - \frac{1}{4 \left(\frac{\sqrt{3} x}{3} - 1\right)}\right)}{9} & \text{for}\: x > - \sqrt{3} \wedge x < \sqrt{3} \end{cases}$$
Simplificar
Gráfica
Trazar el gráfico f(x)
Respuesta [src] 
       ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
1    \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /   \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /
-- - ------------------------------ - ---------------- + ------------------------- + --------------------
12                 36                        36                      36                       36
$$- \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{36} + \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{36} + \frac{1}{12} - \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{36} + \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{36}$$ 
=
= 
       ___ /          /       ___\\     ___    /  ___\     ___ /          /  ___\\     ___    /      ___\
1    \/ 3 *\pi*I + log\-1 + \/ 3 //   \/ 3 *log\\/ 3 /   \/ 3 *\pi*I + log\\/ 3 //   \/ 3 *log\1 + \/ 3 /
-- - ------------------------------ - ---------------- + ------------------------- + --------------------
12                 36                        36                      36                       36
$$- \frac{\sqrt{3} \log{\left(\sqrt{3} \right)}}{36} + \frac{\sqrt{3} \log{\left(1 + \sqrt{3} \right)}}{36} + \frac{1}{12} - \frac{\sqrt{3} \left(\log{\left(-1 + \sqrt{3} \right)} + i \pi\right)}{36} + \frac{\sqrt{3} \left(\log{\left(\sqrt{3} \right)} + i \pi\right)}{36}$$ 
1/12 - sqrt(3)*(pi*i + log(-1 + sqrt(3)))/36 - sqrt(3)*log(sqrt(3))/36 + sqrt(3)*(pi*i + log(sqrt(3)))/36 + sqrt(3)*log(1 + sqrt(3))/36
Respuesta numérica [src] 
0.146695499691746
0.146695499691746

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

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