Integral of $$$- 32 x - 2$$$

Find $$$\int \left(- 32 x - 2\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(- 32 x - 2\right)d x}}} = {\color{red}{\left(- \int{2 d x} - \int{32 x d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=2$$$:

$$- \int{32 x d x} - {\color{red}{\int{2 d x}}} = - \int{32 x d x} - {\color{red}{\left(2 x\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=32$$$ and $$$f{\left(x \right)} = x$$$:

$$- 2 x - {\color{red}{\int{32 x d x}}} = - 2 x - {\color{red}{\left(32 \int{x d x}\right)}}$$

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

$$- 2 x - 32 {\color{red}{\int{x d x}}}=- 2 x - 32 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 2 x - 32 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{\left(- 32 x - 2\right)d x} = - 16 x^{2} - 2 x$$

Simplify:

$$\int{\left(- 32 x - 2\right)d x} = 2 x \left(- 8 x - 1\right)$$

Add the constant of integration:

$$\int{\left(- 32 x - 2\right)d x} = 2 x \left(- 8 x - 1\right)+C$$

$$$\int \left(- 32 x - 2\right)\, dx = 2 x \left(- 8 x - 1\right) + C$$$A