Change of variable to calculate expected value

Question

$X$ and $W$ are independent random variables. $$ Z=X+W $$ $$ W \sim \mathcal{N}(0,\sigma) $$ $$ E[X]=\bar{x} $$ I want to calculate $E[Z]$ with respect to the joint pdf $p(z,x)$ $$ E[Z]=\int\int (x+w)p(x,z)dxdz = \bar{x} + \int\int wp(x,z)dxdz $$ How to calculate $$ \int\int wp(x,z)dxdz $$ ?