TITLE: Graphs for which a calculus student can reasonably compute the arclength
QUESTION [63 upvotes]: Given a differentiable real-valued function $f$, the arclength of its graph on $[a,b]$ is given by 
$$\int_a^b\sqrt{1+\left(f'(x)\right)^2}\,\mathrm{d}x$$
For many choices of $f$ this can be a tricky integral to evaluate, especially for calculus students first learning integration. I've found a few choices of $f$ that make the computation pretty easy: 

Letting $f$ be linear is super easy, but then you don't even need the formula.
Taking $f$ of the form $(\text{stuff})^{\frac{3}{2}}$ might work out nicely if $\text{stuff}$ is chosen carefully.
Calculating it for $f(x) = \sqrt{1-x^2}$ is alright if you remember that $\int\frac{1}{x^2+1}\,\mathrm{d}x$ is $\arctan(x)+C$. 
Letting $f(x) = \ln(\sec(x))$ results in $\int\sec(x)\,\mathrm{d}x$, which classically sucks.

But it looks like most choices of $f$ suggest at least a trig substitution $f'(x) \mapsto \tan(\theta)$, and will be computationally intensive, and unreasonable to ask a student to do. Are there other examples of a function $f$ such that computing the arclength of the graph of $f$ won't be too arduous to ask a calculus student to do?

REPLY [3 votes]: Let $u(x) = f'(x)$ and $v(x) = \sqrt{1+u(x)^2}$. So we want $u$ to have an elementary integral (so that we can write down $f$ on the assignment sheet) and $v$ to have an elementary integral (so our students can solve it.) In other words, we want functions $u$ and $v$, both with elementary integrals, so that $v^2 = 1 + u^2$.
Rewrite this as $(v+u) (v-u) = 1$. If $v$ and $u$ have elementary integrals then so do $v+u$ and $v-u$. Conversely, $v = \tfrac{1}{2} \left( (v+u) + (v-u) \right)$ and $u = \tfrac{1}{2} \left( (v+u) - (v-u) \right)$ so, if $v \pm u$ have elementary integrals, then so do $u$ and $v$. So the problem reduces to finding $h$ where $h$ and $1/h$ both have elementary integrals.
Some candidates for $h$ from this perspective:

Any rational function.
Any rational funtion of $e^x$.
Any rational function of $\sin x$ and $\cos x$.