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For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$ In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...

Let $$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$ where $y\in(0,\infty)$, $u\in(0,1)$, and $$f(t):=t+\pi (1-t) t \cot (\pi t).$$ Here are the graphs of $f$ (black), ...