Evaluate the integral.

${\int }_2^8\frac{x^{\frac{7}{3}}-x^{\frac{1}{3}}}{x^{\frac{4}{3}}}dx$

Let $h$ be a function symmetric about $x=5$. Which of the following is correct?

Determine the arc length of the curve $y=(2x+1{)}^{\frac{3}{2}}$ from $x=0$ to $x=c$.

A cyclist's velocity (in $\text{mi/hr}$) is described by the following piecewise function for a $2$-hour ride. Find the function that gives the displacement of the cyclist over the interval $[0,t]$, where $0\le t\le 2$.

What is the displacement function $s\left(t\right)$?

Compute ${\int }_{-\frac{\pi }{4}}^{\frac{\pi }{4}}(\cos 6x+\sin x\cos x-4{\sin }^{3}x)dx$.