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

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

Integrate term by term:

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

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

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

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

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

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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