Integral of $$$a t$$$ with respect to $$$t$$$

Find $$$\int a t\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=a$$$ and $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{a t d t}}} = {\color{red}{a \int{t d t}}}$$

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

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

Therefore,

$$\int{a t d t} = \frac{a t^{2}}{2}$$

Add the constant of integration:

$$\int{a t d t} = \frac{a t^{2}}{2}+C$$

$$$\int a t\, dt = \frac{a t^{2}}{2} + C$$$A