Re: 22/7 - pi integral and a sci.math challenge

On Fri, 07 Aug 2009 10:52:53 -0700, Dave L. Renfro wrote:

There is a neat folklore definite integral whose value is 22/7 - pi:

The integral from x = 0 to x = 1 of

(x^4)*(1 - x)^4 / (1 + x^2)

is equal to 22/7 - pi.

The evaluation by hand is much simpler than it looks, or at least it's
much simpler than I thought it would be when I once decided to see how
hard it'd be to do it by hand. For those interested, the details are
posted here:

ap-calculus -- something for pi day (11 March 2008)
http://mathforum.org/kb/message.jspa?messageID=6133376

See also:

http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80

The other day I came across Problem #557 in "Pi Mu Epsilon Journal"
[Volume 7, Number 9, Fall 1983, p. 615], which is:

"It is known and easy to show with elementary calculus that [the
integral above] = 22/7 - pi. Find a definite integral whose value is
193/71 - e, where e is the base of natural logarithms."

I've since come across some solutions [Vol. 8, No. 1, Fall 1984, pp.
59-60], but all 5 published solutions are extremely contrived:

(a) integrate 1 from x=0 to x = 193/71 - e

(b) integrate 193/71 - e from x=0 to x=1

(c) integrate 122/71 - e^x from x=0 to x=1

(d) integrate e^x from x=0 to x = ln(193/71)

(e) integrate 122x^70 - e^x from x=0 to x=1

At the end is the following editorial note:

"It was hoped that some delightful integral such as the given one for
22/7 - pi would be found. Perhaps some clever reader will still discover
an elegant integral for the desired 193/71 - e."

Thus far I have only gotten up to the 1986 Pi Mu Epsilon Journal issues,
so maybe at some later time something "delightful" was published.
However, I thought I'd throw this out to the sci.math community and see
what you people come up with.

Dave L. Renfro

Note quite what you were asking for, but the integrals int{0 to 1} (x^n/
n!) e^(1-x) dx = e - (1+1+1/2+1/3!+...+1/n!) seem to be of the same
spirit.
.

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