Integral von $$$\left(1 - x^{2}\right)^{\frac{3}{2}}$$$

Bestimme $$$\int \left(1 - x^{2}\right)^{\frac{3}{2}}\, dx$$$.

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

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

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

Daher,

$$$\left(1 - x^{2}\right)^{\frac{3}{2}} = \left(1 - \sin^{2}{\left( u \right)}\right)^{\frac{3}{2}}$$$

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

$$$\left(1 - \sin^{2}{\left( u \right)}\right)^{\frac{3}{2}}=\left(\cos^{2}{\left( u \right)}\right)^{\frac{3}{2}}$$$

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

$$$\left(\cos^{2}{\left( u \right)}\right)^{\frac{3}{2}} = \cos^{3}{\left( u \right)}$$$

Das Integral kann umgeschrieben werden als

$${\color{red}{\int{\left(1 - x^{2}\right)^{\frac{3}{2}} d x}}} = {\color{red}{\int{\cos^{4}{\left(u \right)} d u}}}$$

Wende die Potenzreduktionsformel $$$\cos^{4}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{\cos{\left(4 \alpha \right)}}{8} + \frac{3}{8}$$$ mit $$$\alpha= u $$$ an:

$${\color{red}{\int{\cos^{4}{\left(u \right)} d u}}} = {\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=\frac{1}{8}$$$ und $$$f{\left(u \right)} = 4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3$$$ an:

$${\color{red}{\int{\left(\frac{\cos{\left(2 u \right)}}{2} + \frac{\cos{\left(4 u \right)}}{8} + \frac{3}{8}\right)d u}}} = {\color{red}{\left(\frac{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}{8}\right)}}$$

Gliedweise integrieren:

$$\frac{{\color{red}{\int{\left(4 \cos{\left(2 u \right)} + \cos{\left(4 u \right)} + 3\right)d u}}}}{8} = \frac{{\color{red}{\left(\int{3 d u} + \int{4 \cos{\left(2 u \right)} d u} + \int{\cos{\left(4 u \right)} d u}\right)}}}{8}$$

Wenden Sie die Konstantenregel $$$\int c\, du = c u$$$ mit $$$c=3$$$ an:

$$\frac{\int{4 \cos{\left(2 u \right)} d u}}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{3 d u}}}}{8} = \frac{\int{4 \cos{\left(2 u \right)} d u}}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\left(3 u\right)}}}{8}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=4$$$ und $$$f{\left(u \right)} = \cos{\left(2 u \right)}$$$ an:

$$\frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{4 \cos{\left(2 u \right)} d u}}}}{8} = \frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\left(4 \int{\cos{\left(2 u \right)} d u}\right)}}}{8}$$

Sei $$$v=2 u$$$.

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

Also,

$$\frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{\cos{\left(2 u \right)} d u}}}}{2} = \frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{2}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ mit $$$c=\frac{1}{2}$$$ und $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ an:

$$\frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{2} d v}}}}{2} = \frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{2}\right)}}}{2}$$

Das Integral des Kosinus ist $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$$\frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{4} = \frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{{\color{red}{\sin{\left(v \right)}}}}{4}$$

Zur Erinnerung: $$$v=2 u$$$:

$$\frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{\sin{\left({\color{red}{v}} \right)}}{4} = \frac{3 u}{8} + \frac{\int{\cos{\left(4 u \right)} d u}}{8} + \frac{\sin{\left({\color{red}{\left(2 u\right)}} \right)}}{4}$$

Sei $$$v=4 u$$$.

Dann $$$dv=\left(4 u\right)^{\prime }du = 4 du$$$ (die Schritte sind » zu sehen), und es gilt $$$du = \frac{dv}{4}$$$.

Also,

$$\frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\int{\cos{\left(4 u \right)} d u}}}}{8} = \frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{8}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ mit $$$c=\frac{1}{4}$$$ und $$$f{\left(v \right)} = \cos{\left(v \right)}$$$ an:

$$\frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\int{\frac{\cos{\left(v \right)}}{4} d v}}}}{8} = \frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(v \right)} d v}}{4}\right)}}}{8}$$

Das Integral des Kosinus ist $$$\int{\cos{\left(v \right)} d v} = \sin{\left(v \right)}$$$:

$$\frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\int{\cos{\left(v \right)} d v}}}}{32} = \frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{{\color{red}{\sin{\left(v \right)}}}}{32}$$

Zur Erinnerung: $$$v=4 u$$$:

$$\frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{\sin{\left({\color{red}{v}} \right)}}{32} = \frac{3 u}{8} + \frac{\sin{\left(2 u \right)}}{4} + \frac{\sin{\left({\color{red}{\left(4 u\right)}} \right)}}{32}$$

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

$$\frac{\sin{\left(2 {\color{red}{u}} \right)}}{4} + \frac{\sin{\left(4 {\color{red}{u}} \right)}}{32} + \frac{3 {\color{red}{u}}}{8} = \frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(x \right)}}} \right)}}{4} + \frac{\sin{\left(4 {\color{red}{\operatorname{asin}{\left(x \right)}}} \right)}}{32} + \frac{3 {\color{red}{\operatorname{asin}{\left(x \right)}}}}{8}$$

Daher,

$$\int{\left(1 - x^{2}\right)^{\frac{3}{2}} d x} = \frac{\sin{\left(2 \operatorname{asin}{\left(x \right)} \right)}}{4} + \frac{\sin{\left(4 \operatorname{asin}{\left(x \right)} \right)}}{32} + \frac{3 \operatorname{asin}{\left(x \right)}}{8}$$

Verwenden Sie die Formeln $$$\sin{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\sin{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{1 - \alpha^{2}}$$$, $$$\cos{\left(2 \operatorname{asin}{\left(\alpha \right)} \right)} = 1 - 2 \alpha^{2}$$$, $$$\cos{\left(2 \operatorname{acos}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, $$$\sinh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha^{2} + 1}$$$, $$$\sinh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha \sqrt{\alpha - 1} \sqrt{\alpha + 1}$$$, $$$\cosh{\left(2 \operatorname{asinh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} + 1$$$, $$$\cosh{\left(2 \operatorname{acosh}{\left(\alpha \right)} \right)} = 2 \alpha^{2} - 1$$$, um den Ausdruck zu vereinfachen:

$$\int{\left(1 - x^{2}\right)^{\frac{3}{2}} d x} = \frac{x \sqrt{1 - x^{2}}}{2} + \frac{\sin{\left(4 \operatorname{asin}{\left(x \right)} \right)}}{32} + \frac{3 \operatorname{asin}{\left(x \right)}}{8}$$

Weiter vereinfachen:

$$\int{\left(1 - x^{2}\right)^{\frac{3}{2}} d x} = - \frac{x^{3} \sqrt{1 - x^{2}}}{4} + \frac{5 x \sqrt{1 - x^{2}}}{8} + \frac{3 \operatorname{asin}{\left(x \right)}}{8}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\left(1 - x^{2}\right)^{\frac{3}{2}} d x} = - \frac{x^{3} \sqrt{1 - x^{2}}}{4} + \frac{5 x \sqrt{1 - x^{2}}}{8} + \frac{3 \operatorname{asin}{\left(x \right)}}{8}+C$$

$$$\int \left(1 - x^{2}\right)^{\frac{3}{2}}\, dx = \left(- \frac{x^{3} \sqrt{1 - x^{2}}}{4} + \frac{5 x \sqrt{1 - x^{2}}}{8} + \frac{3 \operatorname{asin}{\left(x \right)}}{8}\right) + C$$$A