How to find $\lim_{R \to \infty} \int_{0}^{R} \frac{\cos(ax)-\cos(bx)}{x} dx?$

Question

How to find the value of

$$\lim_{R \to \infty} \int_{0}^{R} \frac{\cos(ax)-\cos(bx)}{x} dx?$$

Answer 1

HINTS:

Use Frullani's Integral with a slight modification (deform the contour by using Cauchy's Integral Theorem) to find

$$\int_0^\infty \frac{\cos(ax)-\cos(bx)}{x}\,dx=\log(b/a)$$

Alternatively, introduce a new parameter $\alpha>0$ and evaluate the integral

$$I(\alpha)=\int_0^\infty e^{-\alpha x}\frac{\cos(ax)-\cos(bx)}{x}\,dx$$

by differentiating under the integral sign.