Integral of $$$\frac{x^{2}}{3} - 3 x + 1$$$

Find $$$\int \left(\frac{x^{2}}{3} - 3 x + 1\right)\, dx$$$.

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=1$$$:

$$- \int{3 x d x} + \int{\frac{x^{2}}{3} d x} + {\color{red}{\int{1 d x}}} = - \int{3 x d x} + \int{\frac{x^{2}}{3} d x} + {\color{red}{x}}$$

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

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

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

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = x^{2}$$$:

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

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

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

Therefore,

$$\int{\left(\frac{x^{2}}{3} - 3 x + 1\right)d x} = \frac{x^{3}}{9} - \frac{3 x^{2}}{2} + x$$

Simplify:

$$\int{\left(\frac{x^{2}}{3} - 3 x + 1\right)d x} = \frac{x \left(2 x^{2} - 27 x + 18\right)}{18}$$

Add the constant of integration:

$$\int{\left(\frac{x^{2}}{3} - 3 x + 1\right)d x} = \frac{x \left(2 x^{2} - 27 x + 18\right)}{18}+C$$

$$$\int \left(\frac{x^{2}}{3} - 3 x + 1\right)\, dx = \frac{x \left(2 x^{2} - 27 x + 18\right)}{18} + C$$$A