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

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

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

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

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

Daher,

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

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

$$$\sqrt{\cosh^{2}{\left( u \right)} - 1}=\sqrt{\sinh^{2}{\left( u \right)}}$$$

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

$$$\sqrt{\sinh^{2}{\left( u \right)}} = \sinh{\left( u \right)}$$$

Somit,

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

Wende die Potenzreduktionsformel $$$\sinh^{2}{\left(\alpha \right)} = \frac{\cosh{\left(2 \alpha \right)}}{2} - \frac{1}{2}$$$ mit $$$\alpha= u $$$ an:

$${\color{red}{\int{\sinh^{2}{\left(u \right)} d u}}} = {\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}}$$

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

$${\color{red}{\int{\left(\frac{\cosh{\left(2 u \right)}}{2} - \frac{1}{2}\right)d u}}} = {\color{red}{\left(\frac{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}{2}\right)}}$$

Gliedweise integrieren:

$$\frac{{\color{red}{\int{\left(\cosh{\left(2 u \right)} - 1\right)d u}}}}{2} = \frac{{\color{red}{\left(- \int{1 d u} + \int{\cosh{\left(2 u \right)} d u}\right)}}}{2}$$

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

$$\frac{\int{\cosh{\left(2 u \right)} d u}}{2} - \frac{{\color{red}{\int{1 d u}}}}{2} = \frac{\int{\cosh{\left(2 u \right)} d u}}{2} - \frac{{\color{red}{u}}}{2}$$

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

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

Das Integral wird zu

$$- \frac{u}{2} + \frac{{\color{red}{\int{\cosh{\left(2 u \right)} d u}}}}{2} = - \frac{u}{2} + \frac{{\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2}$$

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

$$- \frac{u}{2} + \frac{{\color{red}{\int{\frac{\cosh{\left(v \right)}}{2} d v}}}}{2} = - \frac{u}{2} + \frac{{\color{red}{\left(\frac{\int{\cosh{\left(v \right)} d v}}{2}\right)}}}{2}$$

Das Integral des hyperbolischen Kosinus ist $$$\int{\cosh{\left(v \right)} d v} = \sinh{\left(v \right)}$$$:

$$- \frac{u}{2} + \frac{{\color{red}{\int{\cosh{\left(v \right)} d v}}}}{4} = - \frac{u}{2} + \frac{{\color{red}{\sinh{\left(v \right)}}}}{4}$$

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

$$- \frac{u}{2} + \frac{\sinh{\left({\color{red}{v}} \right)}}{4} = - \frac{u}{2} + \frac{\sinh{\left({\color{red}{\left(2 u\right)}} \right)}}{4}$$

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

$$\frac{\sinh{\left(2 {\color{red}{u}} \right)}}{4} - \frac{{\color{red}{u}}}{2} = \frac{\sinh{\left(2 {\color{red}{\operatorname{acosh}{\left(x \right)}}} \right)}}{4} - \frac{{\color{red}{\operatorname{acosh}{\left(x \right)}}}}{2}$$

Daher,

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

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:

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

Weiter vereinfachen:

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

Fügen Sie die Integrationskonstante hinzu:

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

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