Arc Length: Difficulty With The Integral The question is to find the arc length of a portion of a function.
$$y=\frac{3}{2}x^{2/3}\text{ on }[1,8]$$
I couldn't quite figure out how to evaluate the integral, so I appealed to the solution manual for aid.

I don't quite understand what they did in the 5th step. Could someone perhaps elucidate it for me?

This makes the integral computable, and it is just rewriting the integrand from step 4.  We see the constants multiply to $1$ (the constant outside the integral is $\frac{3}{2}$, the constant inside is $\frac{2}{3}$, and you can pull the constant out.  The only other thing remaining is the $x^{-\frac{1}{3}}$, which we get by simplifying 
$$\sqrt{\frac{1}{x^\frac{2}{3}}} \mbox{ to } \sqrt{x^\frac{-2}{3}}={x^{-\frac{2}{3}}}^\frac{1}{2}=x^\frac{-1}{3}$$
Combining this all together, we see 
$$
\sqrt{\frac{x^\frac{2}{3}+1}{x^\frac{2}{3}}}=\frac{3}{2}\sqrt{x^\frac{2}{3}+1}\frac{2}{3x^\frac{1}{3}}.
$$
I hope this clarifies that step for you.
