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