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

Bepaal $$$\int \frac{a^{x}}{b}\, dx$$$.

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

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

Apply the exponential rule $$$\int{a^{x} d x} = \frac{a^{x}}{\ln{\left(a \right)}}$$$ with $$$a=a$$$:

$$\frac{{\color{red}{\int{a^{x} d x}}}}{b} = \frac{{\color{red}{\frac{a^{x}}{\ln{\left(a \right)}}}}}{b}$$

Dus,

$$\int{\frac{a^{x}}{b} d x} = \frac{a^{x}}{b \ln{\left(a \right)}}$$

Voeg de integratieconstante toe:

$$\int{\frac{a^{x}}{b} d x} = \frac{a^{x}}{b \ln{\left(a \right)}}+C$$

$$$\int \frac{a^{x}}{b}\, dx = \frac{a^{x}}{b \ln\left(a\right)} + C$$$A