# Definite Integral over Bessel Function

Hello I am interested in evaluating the following integral.

p[s_] := Sqrt[1 - (x - s)^2/(ct)^2]
initialWavePulse[x_] := Exp[-x^2/90]

Integrate[initialWavePulse[s]*{BesselJ[0, s]*(p[s]*t/(2 \[Tau])) +
1/p[s] *BesselJ[1, s] *(p[s]*t/(2 \[Tau])) } , {s, x - ct, x + ct}]


When I evaluate this integral, Mathematica just returns the input. Ultimately, I want the functional dependence of x and t as my answer.

Also, when I evaluate a simpler integral, it gives me a conditional expression which makes no sense:

Integrate[BesselJ[0, s], {s, x + ct, x - ct}]


Edit: To give you a little context, this is a part of the general solution to the linearly damped wave equation.

• If Mma returns the input, it means that it cannot solve the task you gave it. Either you should live with it, or you may try to change the statement of problem. For example, in some problems one cannot solve the integral using Integrate, but NIntegrate works. Oct 1, 2014 at 14:36

For the initial integral, why do you have any reason to believe that there is any sort of closed form? As for the simpler integral, if you do:

Assuming[ x > 0 && c > 0 && t > 0,  Integrate[BesselJ[0, s], {s, x + c t, x - c t}]]


It returns:

1/2 (\[Pi] (-c t + x) BesselJ[1, c t - x] StruveH[0,
c t - x] - \[Pi] (c t + x) BesselJ[1, c t + x] StruveH[0,
c t + x] + (c t - x) BesselJ[0,
c t - x] (-2 + \[Pi] StruveH[1, c t - x]) + (c t + x) BesselJ[0,
c t + x] (-2 + \[Pi] StruveH[1, c t + x]))