Re: Calculate a numerical integral with enough precision

*To*: mathgroup at smc.vnet.net*Subject*: [mg114718] Re: Calculate a numerical integral with enough precision*From*: Mark McClure <mcmcclur at unca.edu>*Date*: Tue, 14 Dec 2010 06:54:37 -0500 (EST)

On Sun, Dec 12, 2010 at 5:45 AM, alphatest <iliurarfwpuap at mailinator.com> wrote: > How can we calculate the following integral up to 10-12 decimal places? > integrate exp(sin(1/x)) , x=0..pi/2 I'm not sure why the fabulous new Levin techniques for oscillatory integrands don't work straight away on this integral in V8. They do work for integrands of the form sin^k(x), however. Using that together with the rapidly convergent series for e^x, you get the following. InputForm[Sum[NIntegrate[(Sin[1/x]^k)/k!, {x, 0, Pi/2}], {k, 0, 17}]] 3.101028987751553 In fact, the following yields a closed form expression (in terms of special functions) that should be within machine precision of the exact number. closed = Expand[Sum[Integrate[(Sin[1/x]^k)/k!, {x, 0, Pi/2}], {k, 0, 17}]]; InputForm[N[closed]] 3.101028987753015 So it looks like we've got more than 10 digits. If you want *provably* correct digits, have a look at chapter 1 of The SIAM 100-Digit Challenge. Mark McClure