That certainly makes sense to me now that you explained it. Thanks. However, when I back-substitute that solution into the integral equation and solve the integral numerically, I do not obtain values that are close enough to convince me it's right (I'd expect at least 2 or 3 digit agreement and preferably 6). And the numerical results reported by Mathematica do not change significantly when I up the precision of the numerical calculations. Here is an example for t=1:

Code:

In[230]:=
y[t_] := 1/(t^2 + 1);
i1 = NIntegrate[y[s]/((t - s)^2 + 25) /.
t -> 1, {s, -Infinity, Infinity}]
i2 = N[1/(t^2 + 36) /. t -> 1]
Out[231]= 0.101889
Out[232]= 0.027027

Granted, there is the chance Mathematica is not correctly doing the numerical integration. However Mathematica is usually pretty good at that especially when it does not report any complications such as highly oscillatory or very slow rate of convergence. I've tried other values of t including complex ones and the expressions do not agree.