I was going over calculus homework with my son the other day, who was frustrated because this problem took him three hours to solve:
Find the arclength between 2 and 5 of the function:
$f(x)$ $=$ $x^5 \over 10$ $+$ $1 \over 6x^3$
...which we solved by taking the first derivative and integrating the "square root of d/dx-squared plus one" over the domain. But, he spun his wheels over the complexity of the indefinite integral and became frustrated.
Assuming he took this to someone near graduation who is effective as a math help TA, How much time would that student mentor need to solve this? What kinds of principles and techniques would he teach my son so he could get there in three minutes instead of 180?