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I recently took a practice GRE and I noticed that the integrals on the test were not too advanced or laborious, so long as one noticed the "trick." For instance, the following integral:

$$ \int_{-\pi/4}^{\pi/4} (\cos(t) + \sqrt{1+t^2} \sin(t)^3\cos(t)^3) dt$$.

If one notices that the latter half of the integral is an odd function, the problem is easy. If one does not, then the problem can be quite difficult. Other good examples are provided by this question and this question

This leads me to believe that it is not the best use of time to drill basic integration methods (trig substitution, partial fractions, etc.) in preparation for the GRE (or at least, it is a waste to work on the most laborious use of those methods).

Rather, it seems that the best thing to do is to know basic integration very well and quickly, and to get good at spotting tricks.

I have two questions:

  • To those who have taken the GRE, is my impression of correct that it is better to get at spotting tricks than it is to drill basic methods that would be good for long integrals?

  • Does anybody have a collection of "tricky" problems? That is, problems where the integration is not so bad, so long as one sees the "trick"?

YuiTo Cheng
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msm
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    You are correct. Many questions have a subtle and fast way and an obvious and slow way to approach them. Since this is a timed test, a good score depends on finding the quick approach as often as you can. – Doug M Sep 27 '17 at 23:32
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    Thinking about symmetry in problems should be a well-taught method, and should not be considered trickery. My mantra, particularly in multivariable calculus, was "Exploit symmetry." – Ted Shifrin Sep 27 '17 at 23:33

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This guy is definitely a classic: $$ \int_{-2}^2\sqrt{4-x^2}\mathrm dx $$ which I recall coming up on something GRE related.

From what I remember, for integrals without "obvious" solutions coming from u substitution or parts, there is usually a symmetry or geometry trick involved.

operatorerror
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    I would not view that example as a trick. Identifying $y=\sqrt {4-x^2}$ as the top half of the circle with center $(0,0)$ and radius $2$ should be in your bones at this point. – zhw. Sep 27 '17 at 23:57
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    @zhw. fine, it's not a trick. I interpreted the question to mean "what integrals can I evaluate without actually writing anything down" – operatorerror Sep 28 '17 at 00:01
  • @zhw. I can see how an average student for the GRE might fail to have that easy observation in his bounds. Mathematical problems are either trivial or impossible they say, and that depends on who is looking at the problem (or if we are talking about the RH for that matter :). – Jonatan B. Bastos Sep 28 '17 at 00:08