The problem is given:

$$u_{tt} - u_{xx} = (x^2 -1)e^{-\frac{x^2}{2}}$$

$$u(x,0) = -e^{-\frac{x^2}{2}}, u_t(x,0) = 0$$

I have already got the general solution as followed, but I have trouble solving the integral,

$$\int_{0}^{t} \int_{x-t+s}^{x+t-s} (y^2-1)e^{-\frac{x^2}{2}}dyds$$

and I tried $\Delta$ method, but it seems to make the equation more complex. Professor, could you please give a hint of solving this problem?