<s>[INST] Inequality for an integral involving $ \exp $, $ \sin $ and $ \cos $ Let $ t > 0 $ and $ k \in \{ 0,1,2,\ldots \} $. Does the following inequality hold?

$$
     \int_{k + 1/2}^{k + 3/2}
     \frac{x \sin(2 \pi x)}{1 + 2 e^{2 \pi t} \cos(2 \pi x) + e^{4 \pi t}}
     \mathrm{d}{x}
\leq \frac{1}{2 \pi} \cdot \frac{1}{(1 - e^{2 \pi t})^{2}}.
$$

Such an inequality appears in the study of Selberg $ \zeta $-functions.
 [/INST] The inequality is true, and follows upon integrating by parts.  The integral is 
$$ 
\int_{k+1/2}^{k+3/2} x d\Big( -\frac{\log (1+2 e^{2\pi t} \cos(2\pi x) +e^{4\pi t}}{4\pi e^{2\pi t}} \Big) 
$$
and integration by parts gives
$$
= \frac{1}{4\pi e^{2\pi t}} \int_{k+1/2}^{k+3/2} \log \frac{1+2e^{2\pi t} \cos (2\pi x) + e^{4\pi t}}{1-2e^{2\pi t} +e^{4\pi t}} dx.
$$
Using $\log (1+y) \le y$, the above is 
$$ 
\le \frac{1}{4\pi e^{2\pi t}} \int_{k+1/2}^{k+3/2} \frac{(2+2\cos(2\pi x))e^{2\pi t}}{(1-e^{2\pi t})^2} dx = \frac{1}{2\pi} \frac{1}{(1-e^{2\pi t})^2}. 
$$
</s>