Integral von $$$\sqrt{\frac{1 - t}{t}}$$$

Bestimme $$$\int \sqrt{\frac{1 - t}{t}}\, dt$$$.

Die Eingabe wird umgeschrieben: $$$\int{\sqrt{\frac{1 - t}{t}} d t}=\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t}$$$.

Sei $$$u=\sqrt{t}$$$.

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

Das Integral lässt sich umschreiben als

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

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

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

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

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

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

Der Integrand wird zu

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

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

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

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

$$$\sqrt{\cos^{2}{\left( v \right)}} = \cos{\left( v \right)}$$$

Das Integral kann umgeschrieben werden als

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

Wende die Potenzreduktionsformel $$$\cos^{2}{\left(\alpha \right)} = \frac{\cos{\left(2 \alpha \right)}}{2} + \frac{1}{2}$$$ mit $$$\alpha= v $$$ an:

$$2 {\color{red}{\int{\cos^{2}{\left(v \right)} d v}}} = 2 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}}$$

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

$$2 {\color{red}{\int{\left(\frac{\cos{\left(2 v \right)}}{2} + \frac{1}{2}\right)d v}}} = 2 {\color{red}{\left(\frac{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}{2}\right)}}$$

Gliedweise integrieren:

$${\color{red}{\int{\left(\cos{\left(2 v \right)} + 1\right)d v}}} = {\color{red}{\left(\int{1 d v} + \int{\cos{\left(2 v \right)} d v}\right)}}$$

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

$$\int{\cos{\left(2 v \right)} d v} + {\color{red}{\int{1 d v}}} = \int{\cos{\left(2 v \right)} d v} + {\color{red}{v}}$$

Sei $$$w=2 v$$$.

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

Also,

$$v + {\color{red}{\int{\cos{\left(2 v \right)} d v}}} = v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}}$$

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

$$v + {\color{red}{\int{\frac{\cos{\left(w \right)}}{2} d w}}} = v + {\color{red}{\left(\frac{\int{\cos{\left(w \right)} d w}}{2}\right)}}$$

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

$$v + \frac{{\color{red}{\int{\cos{\left(w \right)} d w}}}}{2} = v + \frac{{\color{red}{\sin{\left(w \right)}}}}{2}$$

Zur Erinnerung: $$$w=2 v$$$:

$$v + \frac{\sin{\left({\color{red}{w}} \right)}}{2} = v + \frac{\sin{\left({\color{red}{\left(2 v\right)}} \right)}}{2}$$

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

$$\frac{\sin{\left(2 {\color{red}{v}} \right)}}{2} + {\color{red}{v}} = \frac{\sin{\left(2 {\color{red}{\operatorname{asin}{\left(u \right)}}} \right)}}{2} + {\color{red}{\operatorname{asin}{\left(u \right)}}}$$

Zur Erinnerung: $$$u=\sqrt{t}$$$:

$$\frac{\sin{\left(2 \operatorname{asin}{\left({\color{red}{u}} \right)} \right)}}{2} + \operatorname{asin}{\left({\color{red}{u}} \right)} = \frac{\sin{\left(2 \operatorname{asin}{\left({\color{red}{\sqrt{t}}} \right)} \right)}}{2} + \operatorname{asin}{\left({\color{red}{\sqrt{t}}} \right)}$$

Daher,

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \frac{\sin{\left(2 \operatorname{asin}{\left(\sqrt{t} \right)} \right)}}{2} + \operatorname{asin}{\left(\sqrt{t} \right)}$$

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:

$$\int{\frac{\sqrt{1 - t}}{\sqrt{t}} d t} = \sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}$$

Fügen Sie die Integrationskonstante hinzu:

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

$$$\int \sqrt{\frac{1 - t}{t}}\, dt = \left(\sqrt{t} \sqrt{1 - t} + \operatorname{asin}{\left(\sqrt{t} \right)}\right) + C$$$A