An Integration Technique You Probably Didn't Know: Reverse Quotient Rule

In this video, we will discuss in detail how to use the Reverse Quotient Rule to find the integral of (-x^2+1)/(x^2+1)^2. This is a useful technique that can be used to find the antiderivative of a ratio of two functions.
Check out more integrals in this playlist: • Integrals Collection
0:00 Intro
0:15 Main Integral
0:30 Quotient Rule for Derivatives
2:40 Reverse Quotient Rule
6:54 Recap
8:23 Final Exercise
#math #algebra #calculus #integral #derivative #integration
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[@wannabeactuary01]

g(x)^2 = x^2 implies g(x) = x is a choice of g(x)
xf' - f = xe^x - e^x
So as f' = f then f = e^x
So a solution to the integral is g(x) = e^x + C where C is the arbitrary constant of integration.
I suppose that begs the question, is there something clever which uses the product rule on f(1/g)?

[@RobolsMath]

I'll try to explore that perspective, but I think that it's a harder path to take because there are plenty other forms that are not as easy to express in the form f'/g - f/g^2.

[@chrisjuravich3398]

Thank you Robols 😊

[@dreadhaisler6054]

How to solve f(x) using differential equations method? I lost it in the deductive reasoning part 😅

[@RobolsMath]

Write it as y' + (-2x/(x^2+1)) y = (-x^2+1)/(x^2+1) which is a first order linear differential equation (FOLDE). Then you can proceed using standard FOLDE methods. See discussion here: tutorial.math.lamar.edu/Classes/DE/Linear.aspx
I think it's gonna be a longer process though.

[@dreadhaisler6054]

​@@RobolsMathI got y/(x²+1) or f(x)/(x²+1), I think I'm still missing some steps haha. Linear in y multiplied by the integrating factor 1/(x²+1). The right side of the general solution is the problem itself which is pretty interesting

[@dreadhaisler6054]

Ohh I rewatched and the solution makes sense to me now

[@RobolsMath]

Great! I fully explained my solution in the video. I thought you just wanted to know the differential equation way as an extra :)

[@djmolina1499]

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[@RobolsMath]

AI Voiceover c/o elevenlabs. They offer free services for 10k characters per month :)
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