Integral of $$$a - x$$$ with respect to $$$x$$$

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

Integrate term by term:

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

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

$$- \int{x d x} + {\color{red}{\int{a d x}}} = - \int{x d x} + {\color{red}{a x}}$$

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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