Integral von $$$\frac{1}{2 \sqrt{1 - x^{2}}}$$$

Bestimme $$$\int \frac{1}{2 \sqrt{1 - x^{2}}}\, dx$$$.

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

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

Das Integral von $$$\frac{1}{\sqrt{1 - x^{2}}}$$$ ist $$$\int{\frac{1}{\sqrt{1 - x^{2}}} d x} = \operatorname{asin}{\left(x \right)}$$$:

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

Daher,

$$\int{\frac{1}{2 \sqrt{1 - x^{2}}} d x} = \frac{\operatorname{asin}{\left(x \right)}}{2}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{1}{2 \sqrt{1 - x^{2}}} d x} = \frac{\operatorname{asin}{\left(x \right)}}{2}+C$$

$$$\int \frac{1}{2 \sqrt{1 - x^{2}}}\, dx = \frac{\operatorname{asin}{\left(x \right)}}{2} + C$$$A