1 / | | ________ | 5 / 2 | x *\/ x + 5 dx | / 0
Integral(x^5*sqrt(x^2 + 5), (x, 0, 1))
TrigSubstitutionRule(theta=_theta, func=sqrt(5)*tan(_theta), rewritten=125*sqrt(5)*tan(_theta)**5/cos(_theta)**3, substep=ConstantTimesRule(constant=125*sqrt(5), other=tan(_theta)**5/cos(_theta)**3, substep=RewriteRule(rewritten=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=AddRule(substeps=[PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), ConstantTimesRule(constant=-2, other=_u**4, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=-2*_u**4, symbol=_u), PowerRule(base=_u, exp=2, context=_u**2, symbol=_u)], context=_u**6 - 2*_u**4 + _u**2, symbol=_u), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)*sec(_theta)**7, symbol=_theta), ConstantTimesRule(constant=-2, other=tan(_theta)*sec(_theta)**5, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), context=-2*tan(_theta)*sec(_theta)**5, symbol=_theta), URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), RewriteRule(rewritten=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, substep=AddRule(substeps=[URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=6, context=_u**6, symbol=_u), context=tan(_theta)*sec(_theta)**7, symbol=_theta), ConstantTimesRule(constant=-2, other=tan(_theta)*sec(_theta)**5, substep=URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=4, context=_u**4, symbol=_u), context=tan(_theta)*sec(_theta)**5, symbol=_theta), context=-2*tan(_theta)*sec(_theta)**5, symbol=_theta), URule(u_var=_u, u_func=sec(_theta), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=tan(_theta)*sec(_theta)**3, symbol=_theta)], context=tan(_theta)*sec(_theta)**7 - 2*tan(_theta)*sec(_theta)**5 + tan(_theta)*sec(_theta)**3, symbol=_theta), context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta)], context=(sec(_theta)**2 - 1)**2*tan(_theta)*sec(_theta)**3, symbol=_theta), context=tan(_theta)**5*sec(_theta)**3, symbol=_theta), context=125*sqrt(5)*tan(_theta)**5/cos(_theta)**3, symbol=_theta), restriction=True, context=x**5*sqrt(x**2 + 5), symbol=x)
Ahora simplificar:
Añadimos la constante de integración:
Respuesta:
/ 5/2 3/2 7/2\ / | / 2\ / 2\ / 2\ | | | | x | | x | | x | | | ________ | 2*|1 + --| |1 + --| |1 + --| | | 5 / 2 ___ | \ 5 / \ 5 / \ 5 / | | x *\/ x + 5 dx = C + 125*\/ 5 *|- ------------- + ----------- + -----------| | \ 5 3 7 / /
___ ___ 200*\/ 5 62*\/ 6 - --------- + -------- 21 7
=
___ ___ 200*\/ 5 62*\/ 6 - --------- + -------- 21 7
-200*sqrt(5)/21 + 62*sqrt(6)/7
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.