Integraal van $$$a^{x} \ln\left(a\right)$$$ met betrekking tot $$$x$$$

Bepaal $$$\int a^{x} \ln\left(a\right)\, dx$$$.

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

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

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

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

Dus,

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

Voeg de integratieconstante toe:

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

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