Integral of $\frac{8}{x}$

Find $\int \frac{8}{x}\, dx$.

Apply the constant multiple rule $\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$ with $c=8$ and $f{\left(x \right)} = \frac{1}{x}$:

$${\color{red}{\int{\frac{8}{x} d x}}} = {\color{red}{\left(8 \int{\frac{1}{x} d x}\right)}}$$

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

$$8 {\color{red}{\int{\frac{1}{x} d x}}} = 8 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Therefore,

$$\int{\frac{8}{x} d x} = 8 \ln{\left(\left|{x}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{8}{x} d x} = 8 \ln{\left(\left|{x}\right| \right)}+C$$

Answer: $\int{\frac{8}{x} d x}=8 \ln{\left(\left|{x}\right| \right)}+C$