Integral of $$$\frac{5}{7} - 3 x$$$

Find $$$\int \left(\frac{5}{7} - 3 x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(\frac{5}{7} - 3 x\right)d x}}} = {\color{red}{\left(\int{\frac{5}{7} d x} - \int{3 x d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=\frac{5}{7}$$$:

$$- \int{3 x d x} + {\color{red}{\int{\frac{5}{7} d x}}} = - \int{3 x d x} + {\color{red}{\left(\frac{5 x}{7}\right)}}$$

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

$$\frac{5 x}{7} - {\color{red}{\int{3 x d x}}} = \frac{5 x}{7} - {\color{red}{\left(3 \int{x d x}\right)}}$$

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

$$\frac{5 x}{7} - 3 {\color{red}{\int{x d x}}}=\frac{5 x}{7} - 3 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{5 x}{7} - 3 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{\left(\frac{5}{7} - 3 x\right)d x} = - \frac{3 x^{2}}{2} + \frac{5 x}{7}$$

Simplify:

$$\int{\left(\frac{5}{7} - 3 x\right)d x} = \frac{x \left(10 - 21 x\right)}{14}$$

Add the constant of integration:

$$\int{\left(\frac{5}{7} - 3 x\right)d x} = \frac{x \left(10 - 21 x\right)}{14}+C$$

$$$\int \left(\frac{5}{7} - 3 x\right)\, dx = \frac{x \left(10 - 21 x\right)}{14} + C$$$A