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Integraal van $$$\frac{68}{r}$$$
Gerelateerde rekenmachine: Rekenmachine voor bepaalde en oneigenlijke integralen
Uw invoer
Bepaal $$$\int \frac{68}{r}\, dr$$$.
Oplossing
Pas de constante-veelvoudregel $$$\int c f{\left(r \right)}\, dr = c \int f{\left(r \right)}\, dr$$$ toe met $$$c=68$$$ en $$$f{\left(r \right)} = \frac{1}{r}$$$:
$${\color{red}{\int{\frac{68}{r} d r}}} = {\color{red}{\left(68 \int{\frac{1}{r} d r}\right)}}$$
De integraal van $$$\frac{1}{r}$$$ is $$$\int{\frac{1}{r} d r} = \ln{\left(\left|{r}\right| \right)}$$$:
$$68 {\color{red}{\int{\frac{1}{r} d r}}} = 68 {\color{red}{\ln{\left(\left|{r}\right| \right)}}}$$
Dus,
$$\int{\frac{68}{r} d r} = 68 \ln{\left(\left|{r}\right| \right)}$$
Voeg de integratieconstante toe:
$$\int{\frac{68}{r} d r} = 68 \ln{\left(\left|{r}\right| \right)}+C$$
Antwoord
$$$\int \frac{68}{r}\, dr = 68 \ln\left(\left|{r}\right|\right) + C$$$A