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Another Integral *July 25, 2009*

*Posted by Dennis in Problem-solving.*

Tags: Integral, Putnam

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Tags: Integral, Putnam

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This is a problem from the Putnam competition I saw two years ago as a freshman, and I did it during my stats class this year. It uses symmetry in a similar way to Akhil’s post.

Find .

Let , then

and

Also, we have the identity

So

The differentiating-under-the-integral-sign solution here is also quite nice: let . Then

and after a routine computation with partial fractions one finds

hence, rather miraculously after integrating and simplifying,

The tangent substitution is the first official solution. The second official solution uses the substitution instead, which is closely related. The solution I just described is the third official solution, and the fourth is a fairly tedious but non-”magical” solution using series expansion.

Two very clever solutions.

The Wikipedia page on differentiation under the integral sign has a few more examples. As far as I know, however, it isn’t a “standard” trick as normally one uses a contour integral.

a small typo here:

it’s ‘‘，instead of

[...] 其二是 .首先令,则 [...]

liuxiaochuan,

Thanks for catching that, I fixed it.

How “standard” differentiation under the integral is depends on who you ask. As Qiaochu has said, Doron Zeilberger has made something of a theory of it (reference); it is connected with the method of “creative telescoping” which is used in Wilf-Zeilberger algorithms for evaluating hypergeometric sums.

I guess you could say that this method is not standard or fashionable in textbooks currently. I think it should be more widely known.

Yes, that’s more or less what I meant to say; I am not aware of its being widely taught (though I am basing this claim on independent reading of textbooks, which I assume reflect the contents in the corresponding courses), even though it’s useful, as we discussed.