As an addendum to Paul's notes:
The Luck-Stevens formula is useful if you are unwilling or unable to compute function derivatives. This, however, comes at the expense of having to compute a ratio of two integrals.
The Delves-Lyness method,
$$z_0=\oint_\gamma \frac{z\, h^\prime(z)}{h(z)}\mathrm dz,$$
on the other hand, only requires a single integral, but one needs to compute derivatives. If the function whose root is being sought is easily differentiated, this might be acceptable.
Using Paul's examples above, here are the Delves-Lyness equivalents of some of his examples:
(* first root of Bessel function BesselJ[0, u], diamond contour *)
NIntegrate[u (-BesselJ[1, u])/(BesselJ[0, u]), {u, 1, 2 - I, 3, 2 + I, 1}, WorkingPrecision -> 20]/(2 Pi I)
2.4048255576957727688
(* second root of Bessel function BesselJ[0, u], circular contour *)
With[{h = 5, r = 1},
Re[(r/(2 Pi)) NIntegrate[# (-BesselJ[1, #])/(BesselJ[0, #]) &[h + r Exp[I t]] Exp[I t], {t, 0, 2 Pi},
Method -> {"Trapezoidal", "SymbolicProcessing" -> 0},
WorkingPrecision -> 20]]]
5.5200781102863106496
(We use the trapezoidal rule here, as it is very efficient for numerically evaluating such integrands; see e.g. this and this)
(* third root of Bessel function BesselJ[0, u] via FFT *)
With[{h = 9, r = 1, n = 32},
Chop[r Fourier[Table[N[# (-BesselJ[1, #])/(BesselJ[0, #]) &[h + r Exp[I t]], 20],
{t, 0, 2 Pi - Pi/n, Pi/n}],
FourierParameters -> {-1, 1}]][[2]]]
8.653727912911012217
As for the original "goat problem" solution, we have the following:
Re[NIntegrate[# (# Sin[#])/(Sin[#] - # Cos[#] - Pi/2) &[Pi/2 + Pi Exp[I t]/4] Exp[I t],
{t, 0, 2 Pi}, Method -> {"Trapezoidal", "SymbolicProcessing" -> 0},
WorkingPrecision -> 20]/8]
1.9056957293098838949
The survey paper of Austin, Kravanja, and Trefethen is of interest if you wish to delve further into these matters.