Integraal van $$$\frac{a}{b}$$$ met betrekking tot $$$a$$$

Bepaal $$$\int \frac{a}{b}\, da$$$.

Pas de constante-veelvoudregel $$$\int c f{\left(a \right)}\, da = c \int f{\left(a \right)}\, da$$$ toe met $$$c=\frac{1}{b}$$$ en $$$f{\left(a \right)} = a$$$:

$${\color{red}{\int{\frac{a}{b} d a}}} = {\color{red}{\frac{\int{a d a}}{b}}}$$

Pas de machtsregel $$$\int a^{n}\, da = \frac{a^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ toe met $$$n=1$$$:

$$\frac{{\color{red}{\int{a d a}}}}{b}=\frac{{\color{red}{\frac{a^{1 + 1}}{1 + 1}}}}{b}=\frac{{\color{red}{\left(\frac{a^{2}}{2}\right)}}}{b}$$

Dus,

$$\int{\frac{a}{b} d a} = \frac{a^{2}}{2 b}$$

Voeg de integratieconstante toe:

$$\int{\frac{a}{b} d a} = \frac{a^{2}}{2 b}+C$$

$$$\int \frac{a}{b}\, da = \frac{a^{2}}{2 b} + C$$$A