Graphs for which a calculus student can reasonably compute the arclength

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?

Answer

Attribution
Source : Link , Question Author : Mike Pierce , Answer Author : J. M. ain’t a mathematician