1 / | | 4 | y | ----------- dy | ________ | / 2 | \/ 1 - y | / 0
Integral(y^4/sqrt(1 - y^2), (y, 0, 1))
TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**4, substep=RewriteRule(rewritten=(1/2 - cos(2*_theta)/2)**2, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, 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)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta), RewriteRule(rewritten=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, substep=AddRule(substeps=[ConstantTimesRule(constant=1/4, other=cos(2*_theta)**2, substep=RewriteRule(rewritten=cos(4*_theta)/2 + 1/2, substep=AddRule(substeps=[ConstantTimesRule(constant=1/2, 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)/2, symbol=_theta), ConstantRule(constant=1/2, context=1/2, symbol=_theta)], context=cos(4*_theta)/2 + 1/2, symbol=_theta), context=cos(2*_theta)**2, symbol=_theta), context=cos(2*_theta)**2/4, symbol=_theta), ConstantTimesRule(constant=-1/2, other=cos(2*_theta), substep=URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u), symbol=_u), context=cos(2*_theta), symbol=_theta), context=-cos(2*_theta)/2, symbol=_theta), ConstantRule(constant=1/4, context=1/4, symbol=_theta)], context=cos(2*_theta)**2/4 - cos(2*_theta)/2 + 1/4, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta)], context=(1/2 - cos(2*_theta)/2)**2, symbol=_theta), context=sin(_theta)**4, symbol=_theta), restriction=(y > -1) & (y < 1), context=y**4/sqrt(1 - y**2), symbol=y)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
/ | | 4 // ________ ________ \ | y || / 2 / 2 / 2\ | | ----------- dy = C + |<3*asin(y) y*\/ 1 - y y*\/ 1 - y *\1 - 2*y / | | ________ ||--------- - ------------- + ------------------------ for And(y > -1, y < 1)| | / 2 \\ 8 2 8 / | \/ 1 - y | /
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.