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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?

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    $\begingroup$ Precision goal or accuracy goal? It seems to me you want the latter. $\endgroup$
    – Anton Antonov
    Commented Jun 4, 2020 at 23:09
  • $\begingroup$ Presumably any algorithm that can do one can do the other, since the precision is bounded by the error divided by the minimum. But I’d prefer to control the precision, if I have to choose one. $\endgroup$
    – WillG
    Commented Jun 4, 2020 at 23:21

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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?

Both specific integration rules and / or integration strategies can be developed. See the following MSE posts:

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