Integral von $$$\sqrt{4 - 4 \sin^{2}{\left(x \right)}}$$$

Bestimme $$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx$$$.

Den Integranden vereinfachen:

$${\color{red}{\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=2$$$ und $$$f{\left(x \right)} = \sqrt{1 - \sin^{2}{\left(x \right)}}$$$ an:

$${\color{red}{\int{2 \sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\left(2 \int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}\right)}}$$

Dieses Integral (Unvollständiges elliptisches Integral zweiter Art) besitzt keine geschlossene Form:

$$2 {\color{red}{\int{\sqrt{1 - \sin^{2}{\left(x \right)}} d x}}} = 2 {\color{red}{E\left(x\middle| 1\right)}}$$

Daher,

$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\sqrt{4 - 4 \sin^{2}{\left(x \right)}} d x} = 2 E\left(x\middle| 1\right)+C$$

$$$\int \sqrt{4 - 4 \sin^{2}{\left(x \right)}}\, dx = 2 E\left(x\middle| 1\right) + C$$$A