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

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

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

Therefore,

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

Add the constant of integration:

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

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