2 / | | 3 | x | ----------- dx | ________ | / 2 | \/ 4 - x | / 0
Integral(x^3/sqrt(4 - x^2), (x, 0, 2))
TrigSubstitutionRule(theta=_theta, func=2*sin(_theta), rewritten=8*sin(_theta)**3, substep=ConstantTimesRule(constant=8, 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=8*sin(_theta)**3, symbol=_theta), restriction=(x > -2) & (x < 2), context=x**3/sqrt(4 - x**2), symbol=x)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
/ | | 3 // 3/2 \ | x || ________ / 2\ | | ----------- dx = C + |< / 2 \4 - x / | | ________ ||- 4*\/ 4 - x + ----------- for And(x > -2, x < 2)| | / 2 \\ 3 / | \/ 4 - x | /
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.