Let's define an integral
$$\int_{\epsilon}^{\eta} \cos(2\pi Rr |x'\cdot y'|) - \cos(2 \pi R r) \frac{dr}{r} $$
where $0<\epsilon<\eta < \infty$, $R>0$ and $x',y'\in \mathbb{R^n}$ (not really important since $|x'\cdot y'|$ is just a positive real number). This integral comes up in a proof on singular integrals commuting with dilations in Steins book "Singular Integrals and Differentiability Properties of Functions" by Stein on page 40f if anyone is interested.
The claim is that this integral is uniformly bounded in $\epsilon$ and $\eta$ by
$$ C + C\log\frac{1}{|x'\cdot y'|} $$
(where $\log$ is the natural logarithm) "as an integration by part shows" (Stein is known for his notorious understatement of not-so-trivial observations). Can anyone help me out how to see this?