Integral to Infinity of Cosine p x minus Cosine q x over x

Theorem

$\ds \int_0^\infty \frac {\cos p x - \cos q x} x \rd x = \ln \frac q p$

where $p$ and $q$ are strictly positive real numbers.

Proof

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Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Trigonometric Functions: $15.38$