Integral of $$$624 - 312 x$$$

Find $$$\int \left(624 - 312 x\right)\, dx$$$.

Integrate term by term:

$${\color{red}{\int{\left(624 - 312 x\right)d x}}} = {\color{red}{\left(\int{624 d x} - \int{312 x d x}\right)}}$$

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=624$$$:

$$- \int{312 x d x} + {\color{red}{\int{624 d x}}} = - \int{312 x d x} + {\color{red}{\left(624 x\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=312$$$ and $$$f{\left(x \right)} = x$$$:

$$624 x - {\color{red}{\int{312 x d x}}} = 624 x - {\color{red}{\left(312 \int{x d x}\right)}}$$

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

$$624 x - 312 {\color{red}{\int{x d x}}}=624 x - 312 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=624 x - 312 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Therefore,

$$\int{\left(624 - 312 x\right)d x} = - 156 x^{2} + 624 x$$

Simplify:

$$\int{\left(624 - 312 x\right)d x} = 156 x \left(4 - x\right)$$

Add the constant of integration:

$$\int{\left(624 - 312 x\right)d x} = 156 x \left(4 - x\right)+C$$

$$$\int \left(624 - 312 x\right)\, dx = 156 x \left(4 - x\right) + C$$$A