1 / | | 1 | ------------- dx | 4 2 | x - 6*x + 9 | / 0
Integral(1/(x^4 - 6*x^2 + 9), (x, 0, 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)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
// / / ___\ / ___\\ \ || | | 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 /
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\ 1 \/ 3 *\pi*I + log\-1 + \/ 3 // \/ 3 *log\\/ 3 / \/ 3 *\pi*I + log\\/ 3 // \/ 3 *log\1 + \/ 3 / -- - ------------------------------ - ---------------- + ------------------------- + -------------------- 12 36 36 36 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
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
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.