Integral of $\frac{3}{13 x}$

Find $\int \frac{3}{13 x}\, dx$.

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=\frac{3}{13}$ and $f{\left(x \right)} = \frac{1}{x}$:

$${\color{red}{\int{\frac{3}{13 x} d x}}} = {\color{red}{\left(\frac{3 \int{\frac{1}{x} d x}}{13}\right)}}$$

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

$$\frac{3 {\color{red}{\int{\frac{1}{x} d x}}}}{13} = \frac{3 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}}{13}$$

Therefore,

$$\int{\frac{3}{13 x} d x} = \frac{3 \ln{\left(\left|{x}\right| \right)}}{13}$$

Add the constant of integration:

$$\int{\frac{3}{13 x} d x} = \frac{3 \ln{\left(\left|{x}\right| \right)}}{13}+C$$

$\int \frac{3}{13 x}\, dx = \frac{3 \ln\left(\left|{x}\right|\right)}{13} + C$A