Integral of $$$\frac{1}{a^{2} u}$$$ with respect to $$$u$$$

Find $$$\int \frac{1}{a^{2} u}\, du$$$.

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{a^{2}}$$$ and $$$f{\left(u \right)} = \frac{1}{u}$$$:

$${\color{red}{\int{\frac{1}{a^{2} u} d u}}} = {\color{red}{\frac{\int{\frac{1}{u} d u}}{a^{2}}}}$$

The integral of $$$\frac{1}{u}$$$ is $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\frac{{\color{red}{\int{\frac{1}{u} d u}}}}{a^{2}} = \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{a^{2}}$$

Therefore,

$$\int{\frac{1}{a^{2} u} d u} = \frac{\ln{\left(\left|{u}\right| \right)}}{a^{2}}$$

Add the constant of integration:

$$\int{\frac{1}{a^{2} u} d u} = \frac{\ln{\left(\left|{u}\right| \right)}}{a^{2}}+C$$

$$$\int \frac{1}{a^{2} u}\, du = \frac{\ln\left(\left|{u}\right|\right)}{a^{2}} + C$$$A