This answer seems incomplete
In[36]:= Integrate[a t BesselJ[0, a t] BesselJ[0, b t], {t, 0, Infinity}]
Out[36]= ConditionalExpression[0, b >= 0 && Re[a] > b && a == Re[a]]
I think the answer should be DiracDelta[a-b]
see http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/21/02/02/ or http://dlmf.nist.gov/1.17#E13 1.17.13
This is in Mathematica 9.0.1.0. Is this incomplete? Is it fixed in the latest Mathematica
