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I am currently studying for the GRE math subject test, which heavily tests calculus. I've reviewed most of the basic calculus techniques (integration by parts, trig substitutions, etc.) I am now looking for a list or reference for some lesser-known tricks or clever substitutions that are useful in integration. For example, I learned of this trick

$$\int_a^b f(x) \, dx = \int_a^b f(a + b -x) \, dx$$

in the question Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even

I am especially interested in tricks that can be used without an excessive amount of computation, as I believe (or hope?) that these will be what is useful for the GRE.

  • 4
    I have to wonder if the Weierstrass substitution counts as "lesser-known"...2011-10-09
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    I also don't know if this trick for doubly-infinite integrals is well known: $\int_{-\infty}^\infty f(t)\mathrm dt=\frac12\int_{-\infty}^\infty (f(t)+f(-t))\mathrm dt=\int_0^\infty (f(t)+f(-t))\mathrm dt$. It is usual that the last two integrals are more manageable than the first.2011-10-09
  • 0
    @J.M. $\int_{-\infty}^{\infty} f(t) dt = \int_0^{\infty} (f(t) + f(-t)) dt$ might not hold for $f(t) = 2t/(1+t^2)$ because the integral on the left is undefined (works out to $\infty - \infty$) while the one on the right is $0$2011-10-09
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    @Dilip: But the Cauchy principal value of the integral of your function is indeed zero. :)2011-10-09
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    Another one: it is sometimes helpful to express trigonometric/hyperbolic functions in terms of (complex) exponentials; this allows you to readily do things like partial fraction decomposition...2011-10-09
  • 0
    Still another one: too many people forget that integrals can be reversed: $\int_a^b f(t)\mathrm dt=-\int_b^a f(t)\mathrm dt$. Lots of people forget to exploit periodicity in their integrals, too: $3\int_0^{2\pi} \frac{\mathrm dt}{2+\sin\,t}$ is a bit easier than $\int_0^{6\pi} \frac{\mathrm dt}{2+\sin\,t}$.2011-10-09
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    @J.M. Indeed the Cauchy principal value of the integral of $2t/(1+t^2)$ is $0$, but using the Cauchy principal value when the integral itself is indeterminate can be inappropriate in some circumstances. See [here](http://math.stackexchange.com/questions/64651/expected-value-of-a-continuous-random-variable/64669#64669) for a related discussion which also mentions Cauchy but in a different context.2011-10-09
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    Here I have collected a few tips and tricks.. http://folk.ntnu.no/oistes/Diverse/Integral%20Kokeboken.pdf Yeah, wrong language. But math is universal and you can still look up the propositions and theorems =)2014-02-26
  • 0
    From my personal experience on the GRE - the most important tool in your arsenal is using approximation techniques to get proper bounds on the integrals. This can change a 3-4 minute problem into a 30 second one.2014-02-26
  • 0
    IMO it's Feynman's trick (aka differentiating under the integral sign), but that's becoming more popularly taught2016-06-28

8 Answers 8