Integral of $$$8 x^{2} + 5 x - 9$$$

Find $$$\int \left(8 x^{2} + 5 x - 9\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(8 x^{2} + 5 x - 9\right)d x}}} = {\color{red}{\left(- \int{9 d x} + \int{5 x d x} + \int{8 x^{2} d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=9$$$:

$$\int{5 x d x} + \int{8 x^{2} d x} - {\color{red}{\int{9 d x}}} = \int{5 x d x} + \int{8 x^{2} d x} - {\color{red}{\left(9 x\right)}}$$

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

$$- 9 x + \int{8 x^{2} d x} + {\color{red}{\int{5 x d x}}} = - 9 x + \int{8 x^{2} d x} + {\color{red}{\left(5 \int{x d x}\right)}}$$

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

$$- 9 x + \int{8 x^{2} d x} + 5 {\color{red}{\int{x d x}}}=- 9 x + \int{8 x^{2} d x} + 5 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 9 x + \int{8 x^{2} d x} + 5 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=8$$$ and $$$f{\left(x \right)} = x^{2}$$$:

$$\frac{5 x^{2}}{2} - 9 x + {\color{red}{\int{8 x^{2} d x}}} = \frac{5 x^{2}}{2} - 9 x + {\color{red}{\left(8 \int{x^{2} d x}\right)}}$$

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

$$\frac{5 x^{2}}{2} - 9 x + 8 {\color{red}{\int{x^{2} d x}}}=\frac{5 x^{2}}{2} - 9 x + 8 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=\frac{5 x^{2}}{2} - 9 x + 8 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(8 x^{2} + 5 x - 9\right)d x} = \frac{8 x^{3}}{3} + \frac{5 x^{2}}{2} - 9 x$$

Simplify:

$$\int{\left(8 x^{2} + 5 x - 9\right)d x} = \frac{x \left(16 x^{2} + 15 x - 54\right)}{6}$$

Add the constant of integration:

$$\int{\left(8 x^{2} + 5 x - 9\right)d x} = \frac{x \left(16 x^{2} + 15 x - 54\right)}{6}+C$$

$$$\int \left(8 x^{2} + 5 x - 9\right)\, dx = \frac{x \left(16 x^{2} + 15 x - 54\right)}{6} + C$$$A