Integral of $$$a^{x}$$$ with respect to $$$x$$$

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

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

Therefore,

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

Add the constant of integration:

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

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