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

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

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

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

Das Integral wird zu

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

Das Integral von $$$\frac{1}{u}$$$ ist $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

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

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\operatorname{asin}{\left(x \right)}}}}\right| \right)}$$

Daher,

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

Fügen Sie die Integrationskonstante hinzu:

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

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