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:

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

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