Integral de x^3*arccos3x dx

1. Usamos la integración por partes:

   $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

   que $u{\left(x \right)} = \operatorname{acos}{\left(3 x \right)}$ y que $\operatorname{dv}{\left(x \right)} = x^{3}$.

   Entonces $\operatorname{du}{\left(x \right)} = - \frac{3}{\sqrt{1 - 9 x^{2}}}$.

   Para buscar $v{\left(x \right)}$:

   1. Integral $x^{n}$ es $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$:

      $\int x^{3}\, dx = \frac{x^{4}}{4}$

   Ahora resolvemos podintegral.

2. La integral del producto de una función por una constante es la constante por la integral de esta función:

   $\int \left(- \frac{3 x^{4}}{4 \sqrt{1 - 9 x^{2}}}\right)\, dx = - \frac{3 \int \frac{x^{4}}{\sqrt{1 - 9 x^{2}}}\, dx}{4}$

   TrigSubstitutionRule(theta=_theta, func=sin(_theta)/3, rewritten=sin(_theta)**4/243, substep=ConstantTimesRule(constant=1/243, other=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), context=sin(_theta)**4/243, symbol=_theta), restriction=(x > -1/3) & (x < 1/3), context=x**4/sqrt(1 - 9*x**2), symbol=x)

   Por lo tanto, el resultado es: $- \frac{3 \left(\begin{cases} \frac{x \left(1 - 18 x^{2}\right) \sqrt{1 - 9 x^{2}}}{648} - \frac{x \sqrt{1 - 9 x^{2}}}{162} + \frac{\operatorname{asin}{\left(3 x \right)}}{648} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}\right)}{4}$

3. Ahora simplificar:

   $\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}$

4. Añadimos la constante de integración:

   $\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} \frac{x^{4} \operatorname{acos}{\left(3 x \right)}}{4} - \frac{x^{3} \sqrt{1 - 9 x^{2}}}{48} - \frac{x \sqrt{1 - 9 x^{2}}}{288} + \frac{\operatorname{asin}{\left(3 x \right)}}{864} & \text{for}\: x > - \frac{1}{3} \wedge x < \frac{1}{3} \end{cases}+ \mathrm{constant}$