Integral von $$$- \frac{1}{\sqrt{16 - 4 x^{2}}}$$$

Bestimme $$$\int \left(- \frac{1}{\sqrt{16 - 4 x^{2}}}\right)\, dx$$$.

Den Integranden vereinfachen:

$${\color{red}{\int{\left(- \frac{1}{\sqrt{16 - 4 x^{2}}}\right)d x}}} = {\color{red}{\int{\left(- \frac{1}{2 \sqrt{4 - x^{2}}}\right)d x}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=- \frac{1}{2}$$$ und $$$f{\left(x \right)} = \frac{1}{\sqrt{4 - x^{2}}}$$$ an:

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

Sei $$$x=2 \sin{\left(u \right)}$$$.

Dann $$$dx=\left(2 \sin{\left(u \right)}\right)^{\prime }du = 2 \cos{\left(u \right)} du$$$ (die Schritte sind » zu sehen).

Somit folgt, dass $$$u=\operatorname{asin}{\left(\frac{x}{2} \right)}$$$.

Daher,

$$$\frac{1}{\sqrt{4 - x^{2}}} = \frac{1}{\sqrt{4 - 4 \sin^{2}{\left( u \right)}}}$$$

Verwenden Sie die Identität $$$1 - \sin^{2}{\left( u \right)} = \cos^{2}{\left( u \right)}$$$:

$$$\frac{1}{\sqrt{4 - 4 \sin^{2}{\left( u \right)}}}=\frac{1}{2 \sqrt{1 - \sin^{2}{\left( u \right)}}}=\frac{1}{2 \sqrt{\cos^{2}{\left( u \right)}}}$$$

Setzen wir $$$\cos{\left( u \right)} \ge 0$$$ voraus, so erhalten wir Folgendes:

$$$\frac{1}{2 \sqrt{\cos^{2}{\left( u \right)}}} = \frac{1}{2 \cos{\left( u \right)}}$$$

Also,

$$- \frac{{\color{red}{\int{\frac{1}{\sqrt{4 - x^{2}}} d x}}}}{2} = - \frac{{\color{red}{\int{1 d u}}}}{2}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=1$$$ an:

$$- \frac{{\color{red}{\int{1 d u}}}}{2} = - \frac{{\color{red}{u}}}{2}$$

Zur Erinnerung: $$$u=\operatorname{asin}{\left(\frac{x}{2} \right)}$$$:

$$- \frac{{\color{red}{u}}}{2} = - \frac{{\color{red}{\operatorname{asin}{\left(\frac{x}{2} \right)}}}}{2}$$

Daher,

$$\int{\left(- \frac{1}{\sqrt{16 - 4 x^{2}}}\right)d x} = - \frac{\operatorname{asin}{\left(\frac{x}{2} \right)}}{2}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(- \frac{1}{\sqrt{16 - 4 x^{2}}}\right)d x} = - \frac{\operatorname{asin}{\left(\frac{x}{2} \right)}}{2}+C$$

$$$\int \left(- \frac{1}{\sqrt{16 - 4 x^{2}}}\right)\, dx = - \frac{\operatorname{asin}{\left(\frac{x}{2} \right)}}{2} + C$$$A