Integral de xarcsin(sqrt(x)) dx

1. que $u = \sqrt{x}$.

   Luego que $du = \frac{dx}{2 \sqrt{x}}$ y ponemos $2 du$:

   $\int 2 u^{3} \operatorname{asin}{\left(u \right)}\, du$

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

      $\int u^{3} \operatorname{asin}{\left(u \right)}\, du = 2 \int u^{3} \operatorname{asin}{\left(u \right)}\, du$

      1. Usamos la integración por partes:

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

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

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

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

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

            $\int u^{3}\, du = \frac{u^{4}}{4}$

         Ahora resolvemos podintegral.

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

         $\int \frac{u^{4}}{4 \sqrt{1 - u^{2}}}\, du = \frac{\int \frac{u^{4}}{\sqrt{1 - u^{2}}}\, du}{4}$

         TrigSubstitutionRule(theta=_theta, func=sin(_theta), rewritten=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), restriction=(_u > -1) & (_u < 1), context=_u**4/sqrt(1 - _u**2), symbol=_u)

         Por lo tanto, el resultado es: $\frac{\begin{cases} \frac{u \left(1 - 2 u^{2}\right) \sqrt{1 - u^{2}}}{8} - \frac{u \sqrt{1 - u^{2}}}{2} + \frac{3 \operatorname{asin}{\left(u \right)}}{8} & \text{for}\: u > -1 \wedge u < 1 \end{cases}}{4}$

      Por lo tanto, el resultado es: $\frac{u^{4} \operatorname{asin}{\left(u \right)}}{2} - \frac{\begin{cases} \frac{u \left(1 - 2 u^{2}\right) \sqrt{1 - u^{2}}}{8} - \frac{u \sqrt{1 - u^{2}}}{2} + \frac{3 \operatorname{asin}{\left(u \right)}}{8} & \text{for}\: u > -1 \wedge u < 1 \end{cases}}{2}$

   Si ahora sustituir $u$ más en:

   $\frac{x^{2} \operatorname{asin}{\left(\sqrt{x} \right)}}{2} - \frac{\begin{cases} \frac{\sqrt{x} \left(1 - 2 x\right) \sqrt{1 - x}}{8} - \frac{\sqrt{x} \sqrt{1 - x}}{2} + \frac{3 \operatorname{asin}{\left(\sqrt{x} \right)}}{8} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}}{2}$

2. Ahora simplificar:

   $\begin{cases} \frac{2 x^{\frac{3}{2}} \sqrt{1 - x} + 3 \sqrt{x} \sqrt{1 - x} + 8 x^{2} \operatorname{asin}{\left(\sqrt{x} \right)} - 3 \operatorname{asin}{\left(\sqrt{x} \right)}}{16} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}$

3. Añadimos la constante de integración:

   $\begin{cases} \frac{2 x^{\frac{3}{2}} \sqrt{1 - x} + 3 \sqrt{x} \sqrt{1 - x} + 8 x^{2} \operatorname{asin}{\left(\sqrt{x} \right)} - 3 \operatorname{asin}{\left(\sqrt{x} \right)}}{16} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}+ \mathrm{constant}$

Respuesta:

$\begin{cases} \frac{2 x^{\frac{3}{2}} \sqrt{1 - x} + 3 \sqrt{x} \sqrt{1 - x} + 8 x^{2} \operatorname{asin}{\left(\sqrt{x} \right)} - 3 \operatorname{asin}{\left(\sqrt{x} \right)}}{16} & \text{for}\: x \geq 0 \wedge x < 1 \end{cases}+ \mathrm{constant}$