Integral of $$$3 t^{5}$$$

Find $$$\int 3 t^{5}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=3$$$ and $$$f{\left(t \right)} = t^{5}$$$:

$${\color{red}{\int{3 t^{5} d t}}} = {\color{red}{\left(3 \int{t^{5} d t}\right)}}$$

Apply the power rule $$$\int t^{n}\, dt = \frac{t^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=5$$$:

$$3 {\color{red}{\int{t^{5} d t}}}=3 {\color{red}{\frac{t^{1 + 5}}{1 + 5}}}=3 {\color{red}{\left(\frac{t^{6}}{6}\right)}}$$

Therefore,

$$\int{3 t^{5} d t} = \frac{t^{6}}{2}$$

Add the constant of integration:

$$\int{3 t^{5} d t} = \frac{t^{6}}{2}+C$$

$$$\int 3 t^{5}\, dt = \frac{t^{6}}{2} + C$$$A