Integral of $$$x^{2} - 6 x + 13$$$

Find $$$\int \left(x^{2} - 6 x + 13\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(x^{2} - 6 x + 13\right)d x}}} = {\color{red}{\left(\int{13 d x} - \int{6 x d x} + \int{x^{2} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=13$$$:

$$- \int{6 x d x} + \int{x^{2} d x} + {\color{red}{\int{13 d x}}} = - \int{6 x d x} + \int{x^{2} d x} + {\color{red}{\left(13 x\right)}}$$

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

$$13 x - \int{6 x d x} + {\color{red}{\int{x^{2} d x}}}=13 x - \int{6 x d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=13 x - \int{6 x d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=6$$$ and $$$f{\left(x \right)} = x$$$:

$$\frac{x^{3}}{3} + 13 x - {\color{red}{\int{6 x d x}}} = \frac{x^{3}}{3} + 13 x - {\color{red}{\left(6 \int{x d x}\right)}}$$

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

$$\frac{x^{3}}{3} + 13 x - 6 {\color{red}{\int{x d x}}}=\frac{x^{3}}{3} + 13 x - 6 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{3}}{3} + 13 x - 6 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{\left(x^{2} - 6 x + 13\right)d x} = \frac{x^{3}}{3} - 3 x^{2} + 13 x$$

Simplify:

$$\int{\left(x^{2} - 6 x + 13\right)d x} = \frac{x \left(x^{2} - 9 x + 39\right)}{3}$$

Add the constant of integration:

$$\int{\left(x^{2} - 6 x + 13\right)d x} = \frac{x \left(x^{2} - 9 x + 39\right)}{3}+C$$

$$$\int \left(x^{2} - 6 x + 13\right)\, dx = \frac{x \left(x^{2} - 9 x + 39\right)}{3} + C$$$A