Suppose $t>0$ and $x>0$ and consider the integral $$ \int_0^t d \tau_1 \int_0^{\tau_1} d \tau_2 \int_{-1}^{1} d \mu \ \delta\left(\mu - \frac{- (t - \tau_1)^2 + \tau_2^2 + x^2}{2 x \tau_2} \right) f( \tau_1, \tau_2, \mu ) $$ where $\delta$ is the Dirac delta function and $f$ is some well-behaved function. From here it is obvious that I should be integrating over $\mu$, which would let me write the integrand as $\to f\left(\tau_1, \tau_2, \frac{- (t - \tau_1)^2 + \tau_2^2 + x^2}{2 x \tau_2} \right)$. However this can obviously only be done when $$ -1 < \frac{- (t - \tau_1)^2 + \tau_2^2 + x^2}{2 x \tau_2} < +1 $$ which I think will effect the possible integration ranges for $\tau_1$ and $\tau_2$. I cannot figure out how to reduce the above inequality in a useful way however.
How would one reduce this inequality into useful bounds for the $\tau_1$ and $\tau_2$ range?