Integral de t^6((1-t^2)^-0.5)dt dx

TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=sin(_theta)**6, substep=RewriteRule(rewritten=(1/2 - cos(2*_theta)/2)**3, substep=AlternativeRule(alternatives=[RewriteRule(rewritten=-cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 - 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=-cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=-3/8, 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=-3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=-cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 - 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**3, symbol=_theta), RewriteRule(rewritten=-cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 - 3*cos(2*_theta)/8 + 1/8, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/8, other=cos(2*_theta)**3, substep=RewriteRule(rewritten=(1 - sin(2*_theta)**2)*cos(2*_theta), substep=AlternativeRule(alternatives=[AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1, substep=AddRule(substeps=[ConstantRule(constant=1/2, context=1/2, symbol=_u), ConstantTimesRule(constant=-1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=-_u**2/2, symbol=_u)], context=1/2 - _u**2/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1, substep=AddRule(substeps=[ConstantTimesRule(constant=-1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=-sin(_u)**2*cos(_u)/2, symbol=_u), ConstantTimesRule(constant=1/2, other=cos(_u), substep=TrigRule(func='cos', arg=_u, context=cos(_u), symbol=_u), context=cos(_u)/2, symbol=_u)], context=-sin(_u)**2*cos(_u)/2 + cos(_u)/2, symbol=_u), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), RewriteRule(rewritten=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), substep=AddRule(substeps=[ConstantTimesRule(constant=-1, other=sin(2*_theta)**2*cos(2*_theta), substep=AlternativeRule(alternatives=[URule(u_var=_u, u_func=sin(2*_theta), constant=1/2, substep=ConstantTimesRule(constant=1/2, other=_u**2, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=_u**2, symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), URule(u_var=_u, u_func=2*_theta, constant=1/2, substep=ConstantTimesRule(constant=1/2, other=sin(_u)**2*cos(_u), substep=URule(u_var=_u, u_func=sin(_u), constant=1, substep=PowerRule(base=_u, exp=2, context=_u**2, symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(_u)**2*cos(_u), symbol=_u), context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta)], context=sin(2*_theta)**2*cos(2*_theta), symbol=_theta), context=-sin(2*_theta)**2*cos(2*_theta), symbol=_theta), 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=-sin(2*_theta)**2*cos(2*_theta) + cos(2*_theta), symbol=_theta), context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta)], context=(1 - sin(2*_theta)**2)*cos(2*_theta), symbol=_theta), context=cos(2*_theta)**3, symbol=_theta), context=-cos(2*_theta)**3/8, symbol=_theta), ConstantTimesRule(constant=3/8, 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=3*cos(2*_theta)**2/8, symbol=_theta), ConstantTimesRule(constant=-3/8, 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=-3*cos(2*_theta)/8, symbol=_theta), ConstantRule(constant=1/8, context=1/8, symbol=_theta)], context=-cos(2*_theta)**3/8 + 3*cos(2*_theta)**2/8 - 3*cos(2*_theta)/8 + 1/8, symbol=_theta), context=(1/2 - cos(2*_theta)/2)**3, symbol=_theta)], context=(1/2 - cos(2*_theta)/2)**3, symbol=_theta), context=sin(_theta)**6, symbol=_theta), restriction=(t > -1) & (t < 1), context=t**6/sqrt(1 - t**2), symbol=t)

1. Ahora simplificar:

   $\begin{cases} - \frac{t^{5} \sqrt{1 - t^{2}}}{6} - \frac{5 t^{3} \sqrt{1 - t^{2}}}{24} - \frac{5 t \sqrt{1 - t^{2}}}{16} + \frac{5 \operatorname{asin}{\left(t \right)}}{16} & \text{for}\: t > -1 \wedge t < 1 \end{cases}$

2. Añadimos la constante de integración:

   $\begin{cases} - \frac{t^{5} \sqrt{1 - t^{2}}}{6} - \frac{5 t^{3} \sqrt{1 - t^{2}}}{24} - \frac{5 t \sqrt{1 - t^{2}}}{16} + \frac{5 \operatorname{asin}{\left(t \right)}}{16} & \text{for}\: t > -1 \wedge t < 1 \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} - \frac{t^{5} \sqrt{1 - t^{2}}}{6} - \frac{5 t^{3} \sqrt{1 - t^{2}}}{24} - \frac{5 t \sqrt{1 - t^{2}}}{16} + \frac{5 \operatorname{asin}{\left(t \right)}}{16} & \text{for}\: t > -1 \wedge t < 1 \end{cases}+ \mathrm{constant}$