Why is the function $\frac{1-cosx}{x}$ Lebesgue-integrable on $\mathbb{R}$?
All I am given is that it is bounded and continuous on the real line. Is that enough to show it's improper Lebesgue-integrable? Shouldn't I find a bound for $\int_{a}^{b}|\frac{1-cosx}{x}|dx$ for all $b\geq a$?
It is not Lebesgue-integrable on $\mathbb R$. In fact it's not hard to show that $$ \int_0^R \frac{1-\cos(x)}{x}\; dx \sim \log R \ \text{as}\ R \to \infty$$