Integral of $$$8 f t^{3}$$$ with respect to $$$t$$$

Find $$$\int 8 f t^{3}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=8 f$$$ and $$$f{\left(t \right)} = t^{3}$$$:

$${\color{red}{\int{8 f t^{3} d t}}} = {\color{red}{\left(8 f \int{t^{3} d t}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=3$$$:

$$8 f {\color{red}{\int{t^{3} d t}}}=8 f {\color{red}{\frac{t^{1 + 3}}{1 + 3}}}=8 f {\color{red}{\left(\frac{t^{4}}{4}\right)}}$$

Therefore,

$$\int{8 f t^{3} d t} = 2 f t^{4}$$

Add the constant of integration:

$$\int{8 f t^{3} d t} = 2 f t^{4}+C$$

$$$\int 8 f t^{3}\, dt = 2 f t^{4} + C$$$A