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

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

Schreiben Sie den Integranden in Bezug auf die Kosekans um.:

$${\color{red}{\int{\frac{1}{\sin^{2}{\left(x \right)}} d x}}} = {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}}$$

Das Integral von $$$\csc^{2}{\left(x \right)}$$$ ist $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:

$${\color{red}{\int{\csc^{2}{\left(x \right)} d x}}} = {\color{red}{\left(- \cot{\left(x \right)}\right)}}$$

Daher,

$$\int{\frac{1}{\sin^{2}{\left(x \right)}} d x} = - \cot{\left(x \right)}$$

Fügen Sie die Integrationskonstante hinzu:

$$\int{\frac{1}{\sin^{2}{\left(x \right)}} d x} = - \cot{\left(x \right)}+C$$

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