Is the arclength of $x\sin(\pi/x)$ in the interval $(0, 1)$ finite?

You can simplify things by noticing

$$\sqrt {1+f'(x)^2} > |f'(x)| = | \cos(\pi/x)/(\pi/x)-\sin(\pi/x)| \ge | \cos(\pi/x)/(\pi/x)|-|\sin(\pi/x)|.$$

Since $\sin(\pi/x)$ is bounded, it suffices to show

$$\int_0^1| \cos(\pi/x)/(\pi/x)|\, dx = \infty.$$

Tags:

Real Analysis

Improper Integrals

Arc Length

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