I want to solve the following equation of an integral valued function:
$Q = \int_{0}^{x_p}f(t_p,x)dx$
for some particular $x_p$ at a fixed time $t_p$, given some known $Q$ and an initial $f(0,x)$. Furthermore, I know the derivative of $f$ with respect to $t$, which is a function of $f$ itself; that is, $\frac{\partial{f}}{\partial{t}} = f(t,x)*g(t,x)$.
How would I go about doing this numerically? I understand that it requires evolving $f$ from time 0 to time $t_p$ and then using a numerical integration technique to solve the equation; what are some specific numerical techniques I could use for both steps?