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

You are not logged in.

#1 2013-01-19 22:12:07

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Integration by Parts

Just finished the draft of Integration by Parts

Criticism/suggestions welcome.

Also: would it be better to give the first example BEFORE saying "to help you remember" and the steps? Just because the first-time reader is probably quite lost without seeing how it works.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Offline

#2 2013-01-19 22:26:30

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,478

Re: Integration by Parts

Hi MIF;

What a nice page. The examples check out. Is there any way to space the diagrams a bit further apart?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

Online

#3 2013-01-19 22:40:39

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,387

Re: Integration by Parts

hi MathsIsFun,

I've always taught this as

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.

Also: would it be better to give the first example BEFORE saying "to help you remember" and the steps?

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.

So, suggestions

(i) consider the alternative formulation of the rule.

(ii) maybe include after some examples a 'why does it work' bit.

(iii) add this example

After one application of the rule the problem gets no simpler.

Apply it again and you get back minus the original integral!

But, now re-arrange 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

Offline

#4 2013-01-20 10:32:49

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Integration by Parts

bob bundy wrote:

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.

Point ii is good

Point iii looks like fun!


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Offline

#5 2013-01-20 19:08:07

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,535

Re: Integration by Parts

OK, posted a new version:  Integration by Parts (use refresh!)

Slightly rearranged, with Bob's example at the end, and a footnote on its relation to Product Rule.

What do you all think now?


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

Offline

#6 2013-01-20 20:04:36

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,387

Re: Integration by Parts

hi MathsIsFun,

Great!  up  Thanks for incorporating my suggestions.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

Board footer

Powered by FluxBB