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