Integral von $$$6 \cot{\left(x \right)} \csc{\left(x \right)}$$$
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Ihre Eingabe
Bestimme $$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx$$$.
Lösung
Wende die Konstantenfaktorregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ mit $$$c=6$$$ und $$$f{\left(x \right)} = \cot{\left(x \right)} \csc{\left(x \right)}$$$ an:
$${\color{red}{\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x}}} = {\color{red}{\left(6 \int{\cot{\left(x \right)} \csc{\left(x \right)} d x}\right)}}$$
Das Integral von $$$\cot{\left(x \right)} \csc{\left(x \right)}$$$ ist $$$\int{\cot{\left(x \right)} \csc{\left(x \right)} d x} = - \csc{\left(x \right)}$$$:
$$6 {\color{red}{\int{\cot{\left(x \right)} \csc{\left(x \right)} d x}}} = 6 {\color{red}{\left(- \csc{\left(x \right)}\right)}}$$
Daher,
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}$$
Fügen Sie die Integrationskonstante hinzu:
$$\int{6 \cot{\left(x \right)} \csc{\left(x \right)} d x} = - 6 \csc{\left(x \right)}+C$$
Antwort
$$$\int 6 \cot{\left(x \right)} \csc{\left(x \right)}\, dx = - 6 \csc{\left(x \right)} + C$$$A