Improper (difficult) integrals at SwissMicros Calculator Forum

[peacecalc]

Hello Valentin,

I tried to integrate your fifth integral from your test-set numericaly:

\[ I = \int_{-1}^{1}\exp\left(x + \sin\left(\exp\left(\exp\left(\exp\left(x + \frac{1}{3}\right)\right)\right)\right)\right)dx \] .

The argument from the sin has a range from 5 to 1.9E19 , so its a wild oscillating story. I tried to find an integrand which has a similar value like this definite integral and I find something like this:

\[ I = \int_{-1}^{1}\exp\left(x\cdot\left(\frac{1+\sqrt{5}}{2}\right)\right)dx \] .

Why I search for this? The reason is simple: I programmed a simple midrule integrator. That program gives the first 4 valid digits of your integral above. The prime integration program gives a value over 3 and the simple program of mine does it better: 2.994... (but with 10^6 calls of function, but prime needs only 4 minutes for this). So the midpoint rule is "blind" for the behavier of the function values. If the oscillating is very wild, the rule will calculate a statistical mean value which is near by the true value.
That is only my assumption. I get the idea, the integrand function behave in a way like the function:
\[ \exp(a\cdot x) \] (the graph shows it clearly, it is an exponential form superimposed with a sinus) and a is the golden ratio (maybe?, it is a guess).

My questions are: from where you know so many valid digits for your test-integral?
and: in some posts you wrote about your own in basic programmed integrator for your HP-71. Do you like to share the code or if you have had already shared it, can you tell us the link?

Thank you very much for test suite in case of numeric integration, I learned a lot, even I failed for getting much more valid digits for your integral number 5. I like this experimental way of doing mathematics, because my mathematical skills, aren't that good, they should be.