Integral of $$$\frac{5 x^{3} - 2}{x}$$$

Find $$$\int \frac{5 x^{3} - 2}{x}\, dx$$$.

Expand the expression:

$${\color{red}{\int{\frac{5 x^{3} - 2}{x} d x}}} = {\color{red}{\int{\left(5 x^{2} - \frac{2}{x}\right)d x}}}$$

Integrate term by term:

$${\color{red}{\int{\left(5 x^{2} - \frac{2}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{2}{x} d x} + \int{5 x^{2} d x}\right)}}$$

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

$$\int{5 x^{2} d x} - {\color{red}{\int{\frac{2}{x} d x}}} = \int{5 x^{2} d x} - {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

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

$$\int{5 x^{2} d x} - 2 {\color{red}{\int{\frac{1}{x} d x}}} = \int{5 x^{2} d x} - 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$- 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{5 x^{2} d x}}} = - 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(5 \int{x^{2} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- 2 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\int{x^{2} d x}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 5 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\frac{5 x^{3} - 2}{x} d x} = \frac{5 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}$$

Add the constant of integration:

$$\int{\frac{5 x^{3} - 2}{x} d x} = \frac{5 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{5 x^{3} - 2}{x}\, dx = \left(\frac{5 x^{3}}{3} - 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A