In Gardiner's Quantum Noise the following integral equality is used (eq 3.3.10, 3.3.14): $$\int_0^{\infty}d\omega \omega\mathrm{coth}\left(\frac{a\omega}{2}\right)\cos(\omega(t-t'))=\int_0^{\infty}d\omega \left[\omega\mathrm{coth}\left(a\omega\right)-\omega\right]\cos(\omega(t-t'))$$ $$+\frac{1}{2}\int_{-\infty}^{\infty}d\omega|\omega|e^{i\omega(t-t')}$$ The aim of this is to write the divergent integral on the left as the sum of convergent and divergent parts.
I have been trying to see where this comes from. So far I have noted that: $$\coth(x)=\coth(2x)+\frac{2}{e^{2x}-e^{-2x}}$$ which lets me write: $$=\int_0^\infty d\omega\left[\omega\coth(a\omega)-\omega\right]\cos(\omega(t-t'))+\int_0^{\infty}d\omega\omega\cos(\omega(t-t'))$$ $$+\int_0^{\infty}d\omega\frac{2\omega}{e^{a\omega}-e^{-a\omega}}\cos(\omega(t-t'))$$ Furthermore we have: $$\int_0^{\infty}d\omega\omega\cos(\omega(t-t'))=\frac{1}{2}\int_{-\infty}^{\infty}d\omega|\omega|\cos(\omega(t-t'))$$ $$\int_0^{\infty}d\omega\frac{2\omega}{e^{a\omega}-e^{-a\omega}}\cos(\omega(t-t'))=\frac{1}{2}\int_{-\infty}^{\infty}d\omega\frac{2\omega}{e^{a\omega}-e^{-a\omega}}\cos(\omega(t-t'))$$ From here though I can't see how to proceed, and am especially confused as to where the complex exponential could possibly have come from. Any ideas at all would be welcome.
\begin{align} \int_{-\infty}^{\infty}d\omega|\omega|\cos(\omega(t-t'))=\int_{-\infty}^{\infty}d\omega|\omega|e^{i\omega(t-t')} \end{align}
because $i \int_{-\infty}^{\infty}d\omega|\omega|sin(\omega(t-t'))=0$, we integrate an odd function over an even interval.
Edit: I don't see why this second integral should vanish mathematically. Is this maybe one of the standard physics arguments that one just absorbs some finite value in the redefinition of a physical quantity?