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