Math Explained in English: Indefinite Integration by Substitution
This is my first attempt to put a symbol-bound bit of math into English, for ease of my own comprehension and amusement of others. It took me something like an hour to parse out the explanation in the textbook because it relied on a dense bunch of nested functions, variables flying about left and right, and so forth. So I’m rewriting it to cut the comprehension time down to about ten minutes or so.
First, you should see the symbol-bound version. This guy’s web page is incredibly good on the subject.
Here’s my summary.
So, you’ve got this function you need to integrate. Suppose it’s kind of messy, like say [imagine an integration symbol here] 20x(x2+5)2 dx. Well, if you can express that as a function (f) of a second function (g), multiplied by the derivative of that second function (or a constant multiple of that derivative), then you can just substitute the variable “u” for that second function, drop the multiplication by the derivative out of your consciousness (changing dx to du to represent it, and keeping any constant multiple), and integrate the resulting much simpler function.
Sooo… in the example I gave, the g function (“u”) would be x2+5. The f function is then u2 (not, shockingly, the band), because that meets the condition described above (the second function being buried within the first function). The derivative of the g function is 2x, which is a tenth of the 20x we conveniently have sitting around collecting dust. So we can express the new function to be integrated as: [integral symbol] u2 (10)du, which, by the constant multiple rule, becomes 10 [integral symbol] u2 du, for which the integral is easy: 10u3/3 + C. Abracadabra!
Take that, textbook!
Math people: did I screw this up? Time to chime in…
Also, who wants to find me a text-sized gif or jpg integral symbol clipart?