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I tried to calculate the following integral, but it returned unevaluated.

In[1]:= Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]
Out[1]= Integrate[BesselJ[0, x - BesselJZero[0, 1]]/x, {x, -Infinity, Infinity}]

Any ideas how to make Mathematica calculate it?

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Try this:

In[1]:= Integrate[BesselJ[0, x]/(x + BesselJZero[0, 1]), {x, -Infinity, Infinity}]
Out[1]= Pi StruveH[0, BesselJZero[0, 1]]
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