Integral of $$$\frac{t^{5}}{2}$$$

Find $$$\int \frac{t^{5}}{2}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(t \right)} = t^{5}$$$:

$${\color{red}{\int{\frac{t^{5}}{2} d t}}} = {\color{red}{\left(\frac{\int{t^{5} d t}}{2}\right)}}$$

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

$$\frac{{\color{red}{\int{t^{5} d t}}}}{2}=\frac{{\color{red}{\frac{t^{1 + 5}}{1 + 5}}}}{2}=\frac{{\color{red}{\left(\frac{t^{6}}{6}\right)}}}{2}$$

Therefore,

$$\int{\frac{t^{5}}{2} d t} = \frac{t^{6}}{12}$$

Add the constant of integration:

$$\int{\frac{t^{5}}{2} d t} = \frac{t^{6}}{12}+C$$

$$$\int \frac{t^{5}}{2}\, dt = \frac{t^{6}}{12} + C$$$A