Integral of $$$1 - a$$$

Find $$$\int \left(1 - a\right)\, da$$$.

Integrate term by term:

$${\color{red}{\int{\left(1 - a\right)d a}}} = {\color{red}{\left(\int{1 d a} - \int{a d a}\right)}}$$

Apply the constant rule $$$\int c\, da = a c$$$ with $$$c=1$$$:

$$- \int{a d a} + {\color{red}{\int{1 d a}}} = - \int{a d a} + {\color{red}{a}}$$

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

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

$$$\int \left(1 - a\right)\, da = \frac{a \left(2 - a\right)}{2} + C$$$A