Why Isn’t There an Inverse Quotient Rule?
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In which I present my paper on the Inverse Quotient Rule. Using a motivating example from Stewart’s Essential Calculus, Early Transcendentals (2nd edition), I derive a Quotient Rule version of Integration by Parts (which is itself derived from the Product Rule), show additional examples, and give my opinion on why the technique is not commonly taught.
Most simple quotients do not have elementary anti-derivatives—in other words, there is no anti-derivative function which can be finitely expressed using any combination of trig functions (and their inverses), exponential/log functions, and algebraic functions. Even so, this technique is useful on certain classes of integrals, such as quotients of trig functions.