I know there are some great posts already about why
PrecisionGoal->n doesn't guarantee the result of
NIntegrate will actually have $n$-digit precision.
However, for a monotone increasing or decreasing function, it is conceivable to integrate numerically and guarantee a certain precision. For each partition of the region of integration, it is easy to compute an absolute maximum and minimum value for the integral (via Riemann sums), and then the error $E$ is bounded by $max-min$. Perhaps there are even more sophisticated techniques to deduce a guaranteed error bound.
Is there a way to tell Mathematica that a function is monotone increasing/decreasing and then insist the the result of
NIntegrate is within a certain guaranteed precision goal?