Let $f(x) = x^{-1}\sin(x^{-1})+\cos(x^{-1})$. How can one prove that $\int f^+ = \infty$?

Question

Hunter mentiones in his notes (p.44) that the function $f(x) := x^{-1}\sin(x^{-1})+\cos(x^{-1})$ lacks a defined Lebesgue integral, which must because both $f^+$ and $f^-$ have infinite integrals. Could someone provide some help as to how to show that $$\int_0^1 f^+$$ is indeed infinite?

Answer 1

For every $k \in \mathbb Z_{> 0}$ and every $x \in \left[\frac2{(4k + 1)\pi}, \frac4{(8k + 1)\pi}\right]$, we have that $x^{-1}\sin(x^{-1}) + \cos(x^{-1})$ is at least $\frac1{\sqrt2}\frac{(8k + 1)\pi}4$; but $\sum_{k = 1}^\infty \frac1{\sqrt2}\frac{(8k + 1)\pi}4\left[\frac4{(8k + 1)\pi} - \frac2{(4k + 1)\pi}\right] = \frac1{2\sqrt2}\sum_{k = 1}^\infty \frac1{4k + 1}$ diverges.