Integral of $$$49 t^{2}$$$

Find $$$\int 49 t^{2}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=49$$$ and $$$f{\left(t \right)} = t^{2}$$$:

$${\color{red}{\int{49 t^{2} d t}}} = {\color{red}{\left(49 \int{t^{2} d t}\right)}}$$

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

$$49 {\color{red}{\int{t^{2} d t}}}=49 {\color{red}{\frac{t^{1 + 2}}{1 + 2}}}=49 {\color{red}{\left(\frac{t^{3}}{3}\right)}}$$

Therefore,

$$\int{49 t^{2} d t} = \frac{49 t^{3}}{3}$$

Add the constant of integration:

$$\int{49 t^{2} d t} = \frac{49 t^{3}}{3}+C$$

$$$\int 49 t^{2}\, dt = \frac{49 t^{3}}{3} + C$$$A