Integral of $$$- 30 t^{2}$$$

Find $$$\int \left(- 30 t^{2}\right)\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=-30$$$ and $$$f{\left(t \right)} = t^{2}$$$:

$${\color{red}{\int{\left(- 30 t^{2}\right)d t}}} = {\color{red}{\left(- 30 \int{t^{2} d t}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=2$$$:

$$- 30 {\color{red}{\int{t^{2} d t}}}=- 30 {\color{red}{\frac{t^{1 + 2}}{1 + 2}}}=- 30 {\color{red}{\left(\frac{t^{3}}{3}\right)}}$$

Therefore,

$$\int{\left(- 30 t^{2}\right)d t} = - 10 t^{3}$$

Add the constant of integration:

$$\int{\left(- 30 t^{2}\right)d t} = - 10 t^{3}+C$$

$$$\int \left(- 30 t^{2}\right)\, dt = - 10 t^{3} + C$$$A