Integraal van $$$\csc^{2}{\left(x \right)} + 1$$$
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Uw invoer
Bepaal $$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx$$$.
Oplossing
Integreer termgewijs:
$${\color{red}{\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x}}} = {\color{red}{\left(\int{1 d x} + \int{\csc^{2}{\left(x \right)} d x}\right)}}$$
Pas de constantenregel $$$\int c\, dx = c x$$$ toe met $$$c=1$$$:
$$\int{\csc^{2}{\left(x \right)} d x} + {\color{red}{\int{1 d x}}} = \int{\csc^{2}{\left(x \right)} d x} + {\color{red}{x}}$$
De integraal van $$$\csc^{2}{\left(x \right)}$$$ is $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$:
$$x + {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}} = x + {\color{red}{\left(- \cot{\left(x \right)}\right)}}$$
Dus,
$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}$$
Voeg de integratieconstante toe:
$$\int{\left(\csc^{2}{\left(x \right)} + 1\right)d x} = x - \cot{\left(x \right)}+C$$
Antwoord
$$$\int \left(\csc^{2}{\left(x \right)} + 1\right)\, dx = \left(x - \cot{\left(x \right)}\right) + C$$$A