Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ ¹ ² ³ °
 

You are not logged in. #1 20130120 21:12:07
Integration by PartsJust finished the draft of Integration by Parts "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #2 20130120 21:26:30
Re: Integration by PartsHi MIF; In mathematics, you don't understand things. You just get used to them. I have the result, but I do not yet know how to get it. All physicists, and a good many quite respectable mathematicians are contemptuous about proof. #3 20130120 21:40:39
Re: Integration by Partshi MathsIsFun, Of course it comes to the same thing, but it avoids having a double integral in the rule, so it looks easier to take in. Also you can easily 'prove' it from the product rule of differentiation. For choosing u and v I say look for a 'u' that gets easier/simpler when you differentiate it and a 'dv/dx' that doesn't get any more complicated when you integrate it. You could mention this when you do the e^x times x example. Good examples .... I especially liked the one that just gets more complicated ... good because it will happen to every student at some time and this shows what to do about it.
Your first example is a good starter. If you adopt my suggestion (i) below, then you could pose the problem, introduce what u and dv/dx might be and show how the rule can provide a solution. I would suggest that you also differentiate the answer to demonstrate that it worked. (And include a comment that doing that is a test for any integral). Then you are reday to state the rule formally. After one application of the rule the problem gets no simpler. Apply it again and you get back minus the original integral! But, now rearrange to make the original once more the subject and you have the solution. I just loved that example when I first met it. Bob You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei #4 20130121 09:32:49
Re: Integration by Parts
I thought about using that, but found I couldn't explain it in a direct way ... I seemed to be saying that you already had to have DONE something (dv/dx) before you could go ahead. So I went for the "here is what you have, and this is what you do with it" approach. "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #5 20130121 18:08:07
Re: Integration by PartsOK, posted a new version: Integration by Parts (use refresh!) "The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  Leon M. Lederman #6 20130121 19:04:36
Re: Integration by Partshi MathsIsFun, You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei 