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

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

Für das Integral $$$\int{\operatorname{acos}{\left(y \right)} d y}$$$ verwenden Sie die partielle Integration $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

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

Dann gilt $$$\operatorname{du}=\left(\operatorname{acos}{\left(y \right)}\right)^{\prime }dy=- \frac{1}{\sqrt{1 - y^{2}}} dy$$$ (Rechenschritte siehe ») und $$$\operatorname{v}=\int{1 d y}=y$$$ (Rechenschritte siehe »).

Das Integral wird zu

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

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

Dann $$$du=\left(1 - y^{2}\right)^{\prime }dy = - 2 y dy$$$ (die Schritte sind » zu sehen), und es gilt $$$y dy = - \frac{du}{2}$$$.

Also,

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

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(u \right)} = \frac{1}{\sqrt{u}}$$$ an:

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

Wenden Sie die Potenzregel $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ mit $$$n=- \frac{1}{2}$$$ an:

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

Zur Erinnerung: $$$u=1 - y^{2}$$$:

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

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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