1) What conditions on the integrand make it integrable over $\mathbb{R}$?
I know if a function is continuous and bounded on a closed interval $[-a,a]$ then this is enough for the function to be integrable on $[-a,a]$. But I'm not so sure if this results extends to $\mathbb{R}$? Perhaps with some type of decay conditions are required?
2) I want to prove
$\int_{-\infty }^{\infty } \frac{(\text{cos}(t)-1)}{ t} \, dt$
is integrable?
Is the following a valid argument. Since the integrand is an odd function, I believe the integral will be $0$ on [-a,a], so
$\int_{-\infty }^{\infty } \frac{(\text{cos}(t)-1)}{ t} \, dt = \lim_{a\rightarrow\infty} \int_{-a}^{a} \frac{(\text{cos}(t)-1)}{ t} \, dt = \lim_{a\rightarrow\infty} 0 = 0 $
Hence since i've shown the integral is zero, it must exist, right? A type of proof by construction, I think.