Integral von $$$\frac{1}{x \sqrt{25 - x^{2}}}$$$

Bestimme $$$\int \frac{1}{x \sqrt{25 - x^{2}}}\, dx$$$.

Sei $$$u=\frac{1}{x}$$$.

Dann $$$du=\left(\frac{1}{x}\right)^{\prime }dx = - \frac{1}{x^{2}} dx$$$ (die Schritte sind » zu sehen), und es gilt $$$\frac{dx}{x^{2}} = - du$$$.

Das Integral wird zu

$${\color{red}{\int{\frac{1}{x \sqrt{25 - x^{2}}} d x}}} = {\color{red}{\int{\left(- \frac{1}{\sqrt{25 u^{2} - 1}}\right)d u}}}$$

Wende die Konstantenfaktorregel $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ mit $$$c=-1$$$ und $$$f{\left(u \right)} = \frac{1}{\sqrt{25 u^{2} - 1}}$$$ an:

$${\color{red}{\int{\left(- \frac{1}{\sqrt{25 u^{2} - 1}}\right)d u}}} = {\color{red}{\left(- \int{\frac{1}{\sqrt{25 u^{2} - 1}} d u}\right)}}$$

Sei $$$u=\frac{\cosh{\left(v \right)}}{5}$$$.

Dann $$$du=\left(\frac{\cosh{\left(v \right)}}{5}\right)^{\prime }dv = \frac{\sinh{\left(v \right)}}{5} dv$$$ (die Schritte sind » zu sehen).

Somit folgt, dass $$$v=\operatorname{acosh}{\left(5 u \right)}$$$.

Also,

$$$\frac{1}{\sqrt{25 u ^{2} - 1}} = \frac{1}{\sqrt{\cosh^{2}{\left( v \right)} - 1}}$$$

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

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

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

$$$\frac{1}{\sqrt{\sinh^{2}{\left( v \right)}}} = \frac{1}{\sinh{\left( v \right)}}$$$

Also,

$$- {\color{red}{\int{\frac{1}{\sqrt{25 u^{2} - 1}} d u}}} = - {\color{red}{\int{\frac{1}{5} d v}}}$$

Wenden Sie die Konstantenregel $$$\int c\, dv = c v$$$ mit $$$c=\frac{1}{5}$$$ an:

$$- {\color{red}{\int{\frac{1}{5} d v}}} = - {\color{red}{\left(\frac{v}{5}\right)}}$$

Zur Erinnerung: $$$v=\operatorname{acosh}{\left(5 u \right)}$$$:

$$- \frac{{\color{red}{v}}}{5} = - \frac{{\color{red}{\operatorname{acosh}{\left(5 u \right)}}}}{5}$$

Zur Erinnerung: $$$u=\frac{1}{x}$$$:

$$- \frac{\operatorname{acosh}{\left(5 {\color{red}{u}} \right)}}{5} = - \frac{\operatorname{acosh}{\left(5 {\color{red}{\frac{1}{x}}} \right)}}{5}$$

Daher,

$$\int{\frac{1}{x \sqrt{25 - x^{2}}} d x} = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{1}{x \sqrt{25 - x^{2}}} d x} = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5}+C$$

$$$\int \frac{1}{x \sqrt{25 - x^{2}}}\, dx = - \frac{\operatorname{acosh}{\left(\frac{5}{x} \right)}}{5} + C$$$A