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

∫baf(x)dx=∫baf(a+b−x)dx

in the question Showing that a∫−af(x)1+exdx=a∫0f(x)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.

**Answer**

I don’t know about “lesser known” but many calculus courses pass over hyperbolic functions. Just as the identity sin2(t)+cos2(t)=1 allows one to deal with 1−x2 terms, the identity cosh2(t)−sinh2(t)=1 allows one to deal with 1+x2 terms.

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