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Integraal van $$$\frac{\csc^{2}{\left(x \right)}}{9}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx$$$.
Oplossing
Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{1}{9}$$$ en $$$f{\left(x \right)} = \csc^{2}{\left(x \right)}$$$:
$${\color{red}{\int{\frac{\csc^{2}{\left(x \right)}}{9} d x}}} = {\color{red}{\left(\frac{\int{\csc^{2}{\left(x \right)} d x}}{9}\right)}}$$
De integraal van $$$\csc^{2}{\left(x \right)}$$$ is $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:
$$\frac{{\color{red}{\int{\csc^{2}{\left(x \right)} d x}}}}{9} = \frac{{\color{red}{\left(- \cot{\left(x \right)}\right)}}}{9}$$
Dus,
$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}$$
Voeg de integratieconstante toe:
$$\int{\frac{\csc^{2}{\left(x \right)}}{9} d x} = - \frac{\cot{\left(x \right)}}{9}+C$$
Antwoord
$$$\int \frac{\csc^{2}{\left(x \right)}}{9}\, dx = - \frac{\cot{\left(x \right)}}{9} + C$$$A