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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.

enter image description here

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

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up vote 2 down vote accepted

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.

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The expression in brackets is precisely what you need for the substitution $u=x^{2/3}+1$ to work.

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