For turbulent flow in a circular conduit, the power$-$law velocity distribution is given by$,\frac{v_x}{v_{\text{m}\text{a}\text{x}}}=(1-\frac{r}{R}{)}^{\frac{1}{n}},$ where $R$ is the radius of the tube. Show that the average velocity of such flow can be represented as $\frac{v_{\text{a}\text{v}\text{g}}}{v_{\text{m}\text{a}\text{x}}}=\frac{2n^2}{(n+1)(2n+1)}$