Integral of $$$12 t$$$

Find $$$\int 12 t\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=12$$$ and $$$f{\left(t \right)} = t$$$:

$${\color{red}{\int{12 t d t}}} = {\color{red}{\left(12 \int{t d t}\right)}}$$

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

$$12 {\color{red}{\int{t d t}}}=12 {\color{red}{\frac{t^{1 + 1}}{1 + 1}}}=12 {\color{red}{\left(\frac{t^{2}}{2}\right)}}$$

Therefore,

$$\int{12 t d t} = 6 t^{2}$$

Add the constant of integration:

$$\int{12 t d t} = 6 t^{2}+C$$

$$$\int 12 t\, dt = 6 t^{2} + C$$$A