Integral of $$$7 t^{4}$$$

Find $$$\int 7 t^{4}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=7$$$ and $$$f{\left(t \right)} = t^{4}$$$:

$${\color{red}{\int{7 t^{4} d t}}} = {\color{red}{\left(7 \int{t^{4} d t}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=4$$$:

$$7 {\color{red}{\int{t^{4} d t}}}=7 {\color{red}{\frac{t^{1 + 4}}{1 + 4}}}=7 {\color{red}{\left(\frac{t^{5}}{5}\right)}}$$

Therefore,

$$\int{7 t^{4} d t} = \frac{7 t^{5}}{5}$$

Add the constant of integration:

$$\int{7 t^{4} d t} = \frac{7 t^{5}}{5}+C$$

$$$\int 7 t^{4}\, dt = \frac{7 t^{5}}{5} + C$$$A