Is $f(x) = \frac{\cos(x)}{x}$ Lebesgue integrable over $E = [0,1]$?

Question

My instructor stated that it is straightforward to show that $f(x) = \frac{\cos(x)}{x}$ is not Lebesgue integrable over $E = [0,1]$.

I know that I'll have to show that $$\int_E \biggl|\frac{\cos(x)}{x} \biggr| = \infty$$ just not sure how to.

Answer 1

$\cos(1) \le \cos(x)$ on $[0,1]$, so

$$\int_{(0,1)} \frac{\cos(1)}{x}\,dx \le \int_{(0,1)}\frac{\cos(x)}{x}\,dx.$$

Can you take it from here?