Definite Integral over Bessel Function

Question

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}]

Please help me go about solving this definite integral! Thank You!

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

Answer 1

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]))