We got some very good response to our last week’s problem from several of our “regular” problem-solvers as well as several others who are “new”. There were solutions that were more “algebraic” than others, some that had a more “trigonometric” flavor to them and some that had a combination of both. All the solutions we received this time were correct and they all deserve to be published, but for the sake of brevity I will post just one.
Solution to POW-5: (due to Animesh Datta, Univ of New Mexico)
Note that the given integral may be written as
Now, we use the substitution 
Finally, we use one last (trigonometric) substitution 
Watch out for the next POW that will be posted by Todd!
Source: I had mentioned earlier that Carl Lira had brought this integral to our attention, and he in turn had found it in the MIT Integration Bee archives. This one was from the year 1994.
Trivia: Four out of the six people who sent correct solutions are either Indians or of Indian origin! Coincidence? 
10 comments
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June 27, 2008 at 3:27 am
Sathya Narayanan
Such integrals are common in Indian High School math curricula..
June 27, 2008 at 4:37 am
Paul Shearer
Who comes up with these integrals?
June 27, 2008 at 6:02 am
Vishal Lama
Paul,
If you meant “who comes up or has the time to come up with such impossible-looking or tricky integrals?”, then I will say that it is rather easy to come up with these! There are at least two quick ways to come up with such integrals. For instance, take the integral that was posed in POW-5. One way to come up with it is to first take the function
, differentiate it and just pose the integral which has the resulting derivative of 
as its integrand! Or, a different way to “generate” the integral is to note that 
is integrable using the usual trigonometric substitution. Now, just substitute for 
any complicated-looking expression (in 
), which in this case happens to be 
, and voila, you have a nice-looking integral to compute!
Of course, in research mathematics, integrals will appear “naturally”, so to speak, and not be concocted “artificially” the way I just showed above.
June 28, 2008 at 1:59 am
Todd Trimble
@ Sathya: Vishal gave me the same explanation when I asked him about this. It makes American high school math curricula seem pale by comparison!
I myself didn’t see the elegant t = x + 1/x trick before hearing it from Vishal (after I showed him and Carl Lira my almost embarrassingly complicated solution). But are Indian high school students trained in solving integrals which involve this very specialized substitution? Or is there some more general methodology at play? (I hope this question doesn’t seem rude.)
June 28, 2008 at 4:18 am
Anonymous
this is just a trick. i remember reading arnold’s ode book. he says the same thing about the convoluted nonlinear differential equations you can get by transforming a linear equations.
what good does it do to be able to do this kind indefinite integrals? i would classify this as useless rote mathematics.
June 28, 2008 at 5:38 am
Vishal Lama
To be fair, I must say I don’t agree completely with Anonymous’s comment made above. Sure, this integral can be evaluated using the “trick” outlined in the solution, but that doesn’t mean that the problem cannot be solved without using the aforesaid trick. We received two other distinct solutions (from several readers) that did not employ any kind of trick. One was purely based on a trigonometric substitution and the other on algebraic methods. I guess I should have published those solutions as well just to show how human “ingenuity” at times comes up with nice solutions even in the absence of knowledge of so-called “tricks”.
More importantly, the ability to compute such integrals should not be underestimated. There are lots of time when even with the help of advanced math software programs, such as Mathematica and Maple, one is unable to compute certain kinds of integrals (both definite and indefinite) without the programs throwing up answers that have in them “non-elementary” expressions that are totally “useless” for our purposes.
June 28, 2008 at 6:41 am
Todd Trimble
I do have some sympathy for what Anonymous said, but “trick” may be an overloaded word here. One man’s trick may be another man’s method, and there may be more to an apparent trick than first meets the eye.
Speaking of Arnol’d: if I recall correctly, he recalls some interesting general things about integrating algebraic functions at the end of his little book Huygens and Barrow, Newton and Hooke. There is a general principle that one can “solve” an integral
(where 
is an algebraic function of 
) if the curve described by the locus 
has genus zero (meaning that if we interpret 
as complex variables and interpret this curve as its closure in the complex projective plane, then the resulting Riemann surface is a sphere). There may be some nonsingularity assumption here. Anyway, my point is that integration of such functions does make contact with non-trivial theory, and is not some isolated backwater. (The interesting claim that Arnol’d makes is that Newton had a heuristic proof of this fact!)
Of course, there is the separate issue of what counts as a closed-form solution. I don’t think the answer has been fixed throughout time; in particular, I think I’ve heard it said that in the 19th century, when elliptic functions were all the rage, one considered an explicit solution using elliptic functions perfectly acceptable (speaking of which, I guess Carl Lira told us that that’s what some computer package told him the answer would look like for this week’s problem).
[Edit: what happened to my avatar?]
June 28, 2008 at 7:24 am
Akhil Ravidas
Substitution x + 1/x is considered as a standard technique.. In fact the textbook i used in my high school had around 10 problems based on the same substitution..
June 28, 2008 at 7:28 am
Sathya Narayanan
@ Todd its just tht we came across some problems in our school curricula with such substitutions . so this one looked familiar… we have some standard substitutions like x+1/x ,x-1/x which have been introduced as some standard substitutions to solve calculus problems… infact most of our high school had calculus in it …
July 2, 2008 at 7:00 pm
Shubhendu Trivedi
Yes it is kind of common especially with guys who hammered some incredible amount of stuff for the IIT-JEE, not common in some state boards though.
I would agree with Anon. to a large extent, i never saw any point in doing such integrals at all my self at all though i used to enjoy them. But i would also agree with Sathya above, there are only a few “standard” substitutions and so it is okay, but they are fun none the less. Solving any problem whether “rote” or “not-rote” gives me a certain joy which is always good.