G prime of f of x, times the derivative of f with respect to X, so we can write that as g prime of f of x. We used to call the art of integration "Algebra and Innovation."Īll remember our good friend the chain rule from differential calculus that tells us that if I were to take the derivative with respect to x of g of f of x, g of, let me write those parentheses a littleīit closer, g of f of x, g of f of x, that this is just going to be equal to the derivative of g with respect to f of On a personal note, I have found that the exercises here at the Khan academy don't spend as much time converting seemingly intractable anti-derivatives into an integrable form as when I was in university - it made integration much more of an art form than differentiation, which was pretty much cook-book math: recognize the form, apply the appropriate recipe. U sub is just one simplification technique of many in our toolbox. Sometimes it is helpful to write out the u sub so as just to avoid careless algebraic errors, however, everyone is different with respect to their personal comfort zone and preferences when choosing what to write out when simplifying expressions and what not. You will find over time, as you do more and more problems, you will be able to do more and more in your head, and that is fine. If you can do all the algebraic manipulation and simplification in your head, then you don't actually need to explicitly write out the the u substitution you performed, though you are actually doing it, that is, recognizing some f and f' within the integrand. This method is intimately related to the chain rule for differentiation, which when applied to anti-derivatives is sometimes called the reverse chain rule. U sub is a method for algebraically simplifying the form of a function so that its anti-derivative can be more easily recognized.
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