Integral de $$$\operatorname{acos}{\left(y \right)}$$$

Halla $$$\int \operatorname{acos}{\left(y \right)}\, dy$$$.

Para la integral $$$\int{\operatorname{acos}{\left(y \right)} d y}$$$, utiliza la integración por partes $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Sean $$$\operatorname{u}=\operatorname{acos}{\left(y \right)}$$$ y $$$\operatorname{dv}=dy$$$.

Entonces $$$\operatorname{du}=\left(\operatorname{acos}{\left(y \right)}\right)^{\prime }dy=- \frac{1}{\sqrt{1 - y^{2}}} dy$$$ (los pasos pueden verse ») y $$$\operatorname{v}=\int{1 d y}=y$$$ (los pasos pueden verse »).

Entonces,

$${\color{red}{\int{\operatorname{acos}{\left(y \right)} d y}}}={\color{red}{\left(\operatorname{acos}{\left(y \right)} \cdot y-\int{y \cdot \left(- \frac{1}{\sqrt{1 - y^{2}}}\right) d y}\right)}}={\color{red}{\left(y \operatorname{acos}{\left(y \right)} - \int{\left(- \frac{y}{\sqrt{1 - y^{2}}}\right)d y}\right)}}$$

Sea $$$u=1 - y^{2}$$$.

Entonces $$$du=\left(1 - y^{2}\right)^{\prime }dy = - 2 y dy$$$ (los pasos pueden verse »), y obtenemos que $$$y dy = - \frac{du}{2}$$$.

La integral se convierte en

$$y \operatorname{acos}{\left(y \right)} - {\color{red}{\int{\left(- \frac{y}{\sqrt{1 - y^{2}}}\right)d y}}} = y \operatorname{acos}{\left(y \right)} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$:

$$y \operatorname{acos}{\left(y \right)} - {\color{red}{\int{\frac{1}{2 \sqrt{u}} d u}}} = y \operatorname{acos}{\left(y \right)} - {\color{red}{\left(\frac{\int{\frac{1}{\sqrt{u}} d u}}{2}\right)}}$$

Aplica la regla de la potencia $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=- \frac{1}{2}$$$:

$$y \operatorname{acos}{\left(y \right)} - \frac{{\color{red}{\int{\frac{1}{\sqrt{u}} d u}}}}{2}=y \operatorname{acos}{\left(y \right)} - \frac{{\color{red}{\int{u^{- \frac{1}{2}} d u}}}}{2}=y \operatorname{acos}{\left(y \right)} - \frac{{\color{red}{\frac{u^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}}}}{2}=y \operatorname{acos}{\left(y \right)} - \frac{{\color{red}{\left(2 u^{\frac{1}{2}}\right)}}}{2}=y \operatorname{acos}{\left(y \right)} - \frac{{\color{red}{\left(2 \sqrt{u}\right)}}}{2}$$

Recordemos que $$$u=1 - y^{2}$$$:

$$y \operatorname{acos}{\left(y \right)} - \sqrt{{\color{red}{u}}} = y \operatorname{acos}{\left(y \right)} - \sqrt{{\color{red}{\left(1 - y^{2}\right)}}}$$

Por lo tanto,

$$\int{\operatorname{acos}{\left(y \right)} d y} = y \operatorname{acos}{\left(y \right)} - \sqrt{1 - y^{2}}$$

Añade la constante de integración:

$$\int{\operatorname{acos}{\left(y \right)} d y} = y \operatorname{acos}{\left(y \right)} - \sqrt{1 - y^{2}}+C$$

$$$\int \operatorname{acos}{\left(y \right)}\, dy = \left(y \operatorname{acos}{\left(y \right)} - \sqrt{1 - y^{2}}\right) + C$$$A