1 / | | 3 | x | ----------- dx | ________ | / 2 | \/ x + 4 | / 0
Integral(x^3/sqrt(x^2 + 4), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=2*tan(_theta), rewritten=8*sin(_theta)**3/cos(_theta)**4, substep=ConstantTimesRule(constant=8, other=sin(_theta)**3/cos(_theta)**4, substep=RewriteRule(rewritten=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=1, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=-(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, substep=ConstantTimesRule(constant=-1, other=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=(_u**2 - 1)/_u**4, substep=RewriteRule(rewritten=_u**(-2) - 1/_u**4, substep=AddRule(substeps=[PowerRule(base=_u, exp=-2, context=_u**(-2), symbol=_u), ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=-1/_u**4, symbol=_u)], context=_u**(-2) - 1/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(_u**2 - 1)/_u**4, symbol=_u), context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=sin(_theta)/cos(_theta)**2 - sin(_theta)/cos(_theta)**4, substep=AddRule(substeps=[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), ConstantTimesRule(constant=-1, other=sin(_theta)/cos(_theta)**4, substep=URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta), context=-sin(_theta)/cos(_theta)**4, symbol=_theta)], context=sin(_theta)/cos(_theta)**2 - sin(_theta)/cos(_theta)**4, symbol=_theta), context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta)], context=(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), context=-(sin(_theta)*cos(_theta)**2 - sin(_theta))/cos(_theta)**4, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), RewriteRule(rewritten=-sin(_theta)/cos(_theta)**2 + sin(_theta)/cos(_theta)**4, 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), URule(u_var=_u, u_func=cos(_theta), constant=-1, substep=ConstantTimesRule(constant=-1, other=_u**(-4), substep=PowerRule(base=_u, exp=-4, context=_u**(-4), symbol=_u), context=_u**(-4), symbol=_u), context=sin(_theta)/cos(_theta)**4, symbol=_theta)], context=-sin(_theta)/cos(_theta)**2 + sin(_theta)/cos(_theta)**4, symbol=_theta), context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta)], context=(1 - cos(_theta)**2)*sin(_theta)/cos(_theta)**4, symbol=_theta), context=sin(_theta)**3/cos(_theta)**4, symbol=_theta), context=8*sin(_theta)**3/cos(_theta)**4, symbol=_theta), restriction=True, context=x**3/sqrt(x**2 + 4), symbol=x)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
/ | 3/2 | 3 ________ / 2\ | x / 2 \4 + x / | ----------- dx = C - 4*\/ 4 + x + ----------- | ________ 3 | / 2 | \/ x + 4 | /
___ 16 7*\/ 5 -- - ------- 3 3
=
___ 16 7*\/ 5 -- - ------- 3 3
16/3 - 7*sqrt(5)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.