Accuracy of Integral with epsilon

[robve]

(10-08-2021 11:28 PM)lrdheat Wrote:  Did not think the integral was remarkable, but was fascinated by how different calculators handled it as written. My Prime in home shows 40.0000000143 (!). Impressed with the WP 34S.

This integral is efficiently integrated with the Tanh-Sinh method (HP forum post). This method is particularly suitable for your function.

I did a quick check with my implementation: for a specified accuracy of eps=10^-4 Tanh-Sinh only performs 27 integrand evaluations to converge to 40 exactly, or at least very close (depends on the floating point format, single/double IEEE 754 or BCD etc.)

The WP 34S uses Tanh-Sinh. That implementation can be improved though. Tanh-Sinh is not considered a general integration method as it may perform quite poorly in certain (known) cases.

- Rob