$\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx$

Question

Let $\lambda,\mu$ be positive constants.
Let $I_{\epsilon,\eta}=\int^{\eta}_{\epsilon} \frac{cos(\lambda x)-cos(\mu x)}{x}dx\:\:\epsilon,\eta >0$.
Show that, by integration by part, $|I_{\epsilon,\eta}|\leq C_1log(\frac{\mu}{\lambda})+C_2$ for some constant $C_1$,$C_2$ which is not depend on $\epsilon$ and $\eta$.
Also, $\lim \limits_{\epsilon \to 0, \eta \to \infty} I_{\epsilon,\eta} = log(\frac{\mu}{\lambda})$
I have tried the integrate $\frac{1}{x}$ first but I can't get an estimate.

Answer 1

$$\begin{align} I(\epsilon,\eta)&=\int_{\epsilon}^{\eta}\frac{\cos \lambda x-\cos \mu x}{x}\,dx\\\\ &=\int_{\epsilon}^{\eta}\int_{\lambda}^{\mu}\sin(xy)\,dy\,dx\\\\ &=\int_{\lambda}^{\mu}\int_{\epsilon}^{\eta}\sin(xy)\,dx\,dy\\\\ &=\int_{\\\lambda}^{\mu}\frac{\cos \epsilon y-\cos \eta y}{y}\,dy \tag 1 \end{align}$$

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For $\mu>\lambda$, we estimate $(1)$ as

$$\left|\int_{\lambda}^{\mu}\frac{\cos \epsilon y-\cos \eta y}{y}\,dy \right|\le\int_{\lambda}^{\mu}\frac{|\cos \epsilon y-\cos \eta y|}{y}\,dy\le 2\log(\mu/\lambda)$$

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To evaluate the limit of $(1)$ as $\epsilon \to 0$ and $\eta \to \infty$ we invoke the Riemann-Lebesgue Lemma. Then, we have

$$\lim_{\epsilon \to 0}\lim_{\eta\to \infty}\int_{\\\lambda}^{\mu}\frac{\cos \epsilon y-\cos \eta y}{y}\,dy=\int_{\lambda}^{\mu}\frac{1}{y}\,dy=\log(\mu/\lambda)$$

and we are done!