How can I evaluate this improper integral? I have a problem: evaluate $$\int_{0}^{\infty}\frac{\cos(x)-\cos(2x)}{x}dx\,.$$ I am told this integral is not an elementary one and that's why I am stuck where to start. Thank you for helping me.
$$\int_a^{\infty}\frac{\cos(2x)}{x}dx = \int_{2a}^{\infty}\frac{\cos(x)}{x}dx$$ and so this integral equals $$\lim_{a\downarrow 0}\int_a^{2a}\frac{\cos(x)}{x}dx$$ which is easily found to be equal to $\log(2)$ using the inequalities $$1-\frac{x^2}{2}\leq\cos(x)\leq 1.$$
You have to use the limited developement of cos(x) and cos(2x).