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