1 / | | 3 | x | ------------- dx | __________ | / 2 | \/ 1 - 2*x | / 0
Integral(x^3/sqrt(1 - 2*x^2), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=sqrt(2)*sin(_theta)/2, rewritten=sin(_theta)**3/4, substep=ConstantTimesRule(constant=1/4, other=sin(_theta)**3, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), ConstantRule(constant=-1, context=-1, symbol=_u)], context=_u**2 - 1, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), RewriteRule(rewritten=-sin(_theta)*cos(_theta)**2 + sin(_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(_theta)*cos(_theta)**2, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(_theta)*cos(_theta)**2, symbol=_theta), context=-sin(_theta)*cos(_theta)**2, symbol=_theta), TrigRule(func='sin', arg=_theta, context=sin(_theta), symbol=_theta)], context=-sin(_theta)*cos(_theta)**2 + sin(_theta), symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta), symbol=_theta), context=sin(_theta)**3, symbol=_theta), context=sin(_theta)**3/4, symbol=_theta), restriction=(x > -sqrt(2)/2) & (x < sqrt(2)/2), context=x**3/sqrt(1 - 2*x**2), symbol=x)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
/ | | 3 // __________ 3/2 \ | x || / 2 / 2\ / ___ ___\| | ------------- dx = C + |< \/ 1 - 2*x \1 - 2*x / | -\/ 2 \/ 2 || | __________ ||- ------------- + ------------- for And|x > -------, x < -----|| | / 2 \\ 4 12 \ 2 2 // | \/ 1 - 2*x | /
1 I - - - 6 3
=
1 I - - - 6 3
1/6 - i/3
(0.143831844407155 - 0.336230285161503j)
(0.143831844407155 - 0.336230285161503j)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.