Integral of $$$\frac{\left(x - 2\right)^{2}}{x}$$$

Find $$$\int \frac{\left(x - 2\right)^{2}}{x}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{\left(x - 2\right)^{2}}{x} d x}}} = {\color{red}{\int{\left(x - 4 + \frac{4}{x}\right)d x}}}$$

Integrate term by term:

$${\color{red}{\int{\left(x - 4 + \frac{4}{x}\right)d x}}} = {\color{red}{\left(- \int{4 d x} + \int{\frac{4}{x} d x} + \int{x d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=4$$$:

$$\int{\frac{4}{x} d x} + \int{x d x} - {\color{red}{\int{4 d x}}} = \int{\frac{4}{x} d x} + \int{x d x} - {\color{red}{\left(4 x\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

$$- 4 x + \int{\frac{4}{x} d x} + {\color{red}{\int{x d x}}}=- 4 x + \int{\frac{4}{x} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 4 x + \int{\frac{4}{x} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:

$$\frac{x^{2}}{2} - 4 x + {\color{red}{\int{\frac{4}{x} d x}}} = \frac{x^{2}}{2} - 4 x + {\color{red}{\left(4 \int{\frac{1}{x} d x}\right)}}$$

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

$$\frac{x^{2}}{2} - 4 x + 4 {\color{red}{\int{\frac{1}{x} d x}}} = \frac{x^{2}}{2} - 4 x + 4 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Therefore,

$$\int{\frac{\left(x - 2\right)^{2}}{x} d x} = \frac{x^{2}}{2} - 4 x + 4 \ln{\left(\left|{x}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{\left(x - 2\right)^{2}}{x} d x} = \frac{x^{2}}{2} - 4 x + 4 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{\left(x - 2\right)^{2}}{x}\, dx = \left(\frac{x^{2}}{2} - 4 x + 4 \ln\left(\left|{x}\right|\right)\right) + C$$$A