How I choose which trig substitution to do

Trig substitution is a fancy kind of substitution used to help find the integral of a particular family of fancy functions. These fancy functions involve things like a² + x² or a² - x² or x² - a² , usually under root signs or inside half-powers, and the purpose of trig substitution is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier. One particular thing the students struggle with is choosing which trig substitution to do. 

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