The last line in the following block shows the join posterior distribution for the mean $\mu$ and precision $\tau$ for a normal distribution:
In order to obtain the marginal posterior for $\tau$ we need to integrate out $\mu$. However, I haven't been able to find a step by step solution.
Hint: Complete the square in the final exponential of your last line until you can write $$ p(\tau,\mu|x)\propto f(\tau)\exp\left\{-\frac{\tau \cdot \rho}{2}(\mu-m)^2\right\}, $$ for some values $\rho$ and $m$, and now use the familiar Gaussian integral.
Can fill in more details if/when you need, but really there isn't much more going on here than completing the square until you have a recognisable Gaussian, with some slightly tedious, but probably worth doing, keeping track of those terms involving $\tau$.