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