Integraal van $$$\frac{5}{7 x}$$$

Bepaal $$$\int \frac{5}{7 x}\, dx$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ toe met $$$c=\frac{5}{7}$$$ en $$$f{\left(x \right)} = \frac{1}{x}$$$:

$${\color{red}{\int{\frac{5}{7 x} d x}}} = {\color{red}{\left(\frac{5 \int{\frac{1}{x} d x}}{7}\right)}}$$

De integraal van $$$\frac{1}{x}$$$ is $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$:

$$\frac{5 {\color{red}{\int{\frac{1}{x} d x}}}}{7} = \frac{5 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{7}$$

Dus,

$$\int{\frac{5}{7 x} d x} = \frac{5 \ln{\left(\left|{x}\right| \right)}}{7}$$

Voeg de integratieconstante toe:

$$\int{\frac{5}{7 x} d x} = \frac{5 \ln{\left(\left|{x}\right| \right)}}{7}+C$$

$$$\int \frac{5}{7 x}\, dx = \frac{5 \ln\left(\left|{x}\right|\right)}{7} + C$$$A