Divergence of $\int_{0}^{+\infty }\frac{\cos x}{x}\ \mathrm dx $

Question

How to prove that $\int_{0}^{+\infty }\frac{\cos x}{x}\ \mathrm dx $ diverges ?

Answer 1

There are two different things that make this integral improper: 1) the fact that $\cos(x)/x \to \infty$ as $x \to 0$, and 2) that the upper limit of integration is $\infty$. We only need to show that one of those causes divergence to say that the integral diverges. I'll focus on #1.

When $0 \leq x < \pi/3$, $\cos x > 1/2$. Therefore $$ \int_0^{\pi/3} \frac{\cos x}{x} \,dx \geq \int_0^{\pi/3} \frac{1}{2x} \,dx $$ Since the integral on the right diverges, the integral on the left must also diverge. So the full integral from $0$ to $\infty$ necessarily also diverges.