Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$ $$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$
for $t\in\Bbb{N}$
How might one represent $a,b$ in terms of $t$?
(Note that $a,b\in \Bbb{N}$)
If possible, could one also provide a proof please? 

Using my newfound knowledge of Mellin Transforms, we can note that if we denote
$$F(s)=\int_0^\infty x^{s-1} \space f(x) \text{ d}x$$
Plugging in $f(x)=\text{csch} x$
One might note that $$\frac{a}{b}=F(s+1)$$
which is known as the result given.
