College Math Teaching

September 13, 2021

Integrals of functions with nice inverses

Filed under: change of variable, derivatives, elementary mathematics, integrals, integration by substitution — collegemathteaching @ 4:49 pm

This idea started as a bit of a joke:

https://twitter.com/lightediand/status/1437181306526371845

Of course, for readers of this blog: easy-peasy. u =\sqrt{tan(x)} \rightarrow u^2 =tan(x) \rightarrow x = arctan(u^2), dx = {2udu \over 1+u^4} so the integral is transformed into \int {2u^2 \over 1+u^4} du and so we’ve entered the realm of rational functions. Ok, ok, there is some work to do.

But for now, notice what is really doing on: we have a function under the radical that has an inverse function (IF we are careful about domains) and said inverse function has a derivative which is a rational function

More shortly: let f(x) be such that {d \over dx} f^{-1}(x) = q(x) then:

\int (f(x))^{1 \over n} dx gets transformed: u^n = f(x) \rightarrow x =f^{-1}(u^n) and then dx = nu^{n-1}q(u^n) and the integral becomes \int n u^n q(u^n) du which is a rational function integral.

Yes, yes, we need to mind domains.

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