Let $u(x, t)$ be the solution to the initial-boundary value problem of the wave equation for x, t > 0: $u_{tt} = 4u_{xx}, u_x(0, t) = 0, u(x, 0) = φ(x) = 2x − 1$ and $u_t(x, 0) = ψ(x) = 2x$ if $x ∈ (2, 10)$ and $0$ if $x ∈ (0, 2] ∪ [10,∞)$.
So I know how to set this equation up using $u(x, t) = 1/2 [φ_{odd}(x + 2t) + φ_{odd}(x − 2t)] + 1/4 \int_{x-2t}^{x+2t} ψ_{odd}(s)ds$.
But I'm not sure on what to use in place of $ψ_{odd}(s)ds$, since the conditions there are mentioned. Like if I have an integral from 0 to 6, what am I supposed to do? Should I split it up into 2 integrals, one with 0 and 2, and the other with 2 and 6?