Talk:Integration by parts

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An alternative notation has the advantage that the factors of the original expression are identified as f and g

Why is that an advantage? It seems arbitrary to say it's better to call them f and g than to call them u and v. Michael Hardy 00:27, 15 Nov 2003 (UTC)

That's not what I meant. The "classic" notation tells you what to do when you are integrating "f(x)g'(x)". The "alternative" integrates "f(x)g(x)". See? -- Tarquin 18:58, 15 Nov 2003 (UTC)

Could we chop of the justification section, since the rule is now justified at the beginning? Also I put the definite integral notation first, because the semantics of the indefinite integral form are a lot less clear. It is still in there since it may still be part of the calculus curriculum. Need to explain bound and free variables.

The distribution comment at the end is informative. Maybe more can be said. The distribution article I think should also be rewritten, having two parts

a 1-dimensional dsitribution theory and
a general theory for open sets and manifolds.

CSTAR 21:15, 11 May 2004 (UTC)

1 A practical note
- 1.1 ILATE rule
2 Multidimensional case
3 Please revise the "recursive formulation" section
4 Tablature Method (possible to add)
5 Failed to parse
6 Q: Recursive formulation

[edit] A practical note

I have removed this section from the end of the article. I think it probably belongs on Wikibooks, being a suggestion for students in second-semester calculus. I also did my best to clean up the writing style somewhat, to eliminate use of second person ("you") and incorrect word usage (e.g. the word "derivate", which may or may not actually mean "derivative", but if it does, it's not standard by any means) - please see my changes here: [1]. Aerion //talk 22:07, 23 Dec 2004 (UTC)

In a general way, when an exponential or trigonometric function appears in the expression, it should be chosen as $v'$ :

$\int f(x) e^{2x}\,dx = \frac{1}{2} f(x) e^{2x} - \frac{1}{2} \int f'(x) e^{2x}\,dx\ \dots$

$\int f(x) \cos(4x)\,dx = \frac{1}{4} f(x) \sin(4x) - \frac{1}{4} \int f'(x) \sin(4x)\,dx\ \dots$

A good way to choose U and dV is the mnemonic "DETAIL". D stands for differential, which is what is to be chosen (dV). The rest of the letters give the order of functions to consider. E stands for exponential. T is for trigonometric function. A stands for algebraic. I is for inverse trigonometric, and finally L is for logarithmic, which is typically a poor choice.

This is especially useful when $f (x)$ is a polynomial, since each consecutive derivative of f(x) is simpler, and eventually, is constant. In general, integration by parts if $f (x)$ is easy to differentiate. Otherwise, the substitution rule may need to be employed.

By contrast, logarithmic or inverse trigonometric functions should be chosen as $u$ :

$\int f(x) \log(3x)\,dx = F(x) \log(3x) - \int F(x) \frac{3}{3x}\,dx\ \dots$

$\int f(x) \arcsin(6x)\,dx = F(x) \arcsin(6x) - \int F(x) \frac{6}{\sqrt{1 - (6x)^2}}\,dx\ \dots$

The objective is to reduce the inverse trigonometric function to a fraction inside the integral. If the derivative contains a radical, trigonometric substitution may be useful.

[edit] ILATE rule

I have removed the text below, which is similar to the section I previously removed, seen above. It doesn't really fit the style of the rest of the article, and ought to be on Wikibooks instead, especially since it is clearly targeted for calculus students. :(Aerion //talk 23:03, 5 Feb 2005 (UTC)'

Alternately a bit handy rule is ILATE rule. This rule helps to decide which function must be used as a substitute for f and which for g. This rule works fine in most cases, making the calculations easier.
KEY:
I = Inverse functions
L = Logarithamic functions
A = Algebric functions
T = Trigonometric functions
E = Exponential functions
So if you get cos(x) and log(x) in the product then, according to the rule take "log(x)" as equivalent to "f" in the equation, while "cos(x)" takes the position of second function.
P.S. Sometimes you need to fiddle around and use the LIATE instead in some cases.

Let's keep the LIATE section (or ILATE, but a quick web search reveals that LIATE is more popular). It's extremely handy for someone looking up integration by parts (as I just did). --Ben Kovitz 12:37, 1 October 2005 (UTC)

That section was written by an anon. I did some formatting on it. I would incline on keeping it, as it is at the very bottom of the article and therefore not interfering with anything else, and it seems that it might be indeed useful. Oleg Alexandrov 01:53, 3 October 2005 (UTC)

It's not at the bottom any longer... Melchoir 08:45, 8 December 2005 (UTC)

[edit] Multidimensional case

I don't have a proof at the moment, but the textbook I'm looking at has this (using $f i, j$ to mean the value of the ith component of the derivative in the j direction of f and the Einstein summation convention).

$\int_\Omega w_i \sigma_{ij,j}\, d\Omega = -\int_\Omega w_{i,j}\sigma_{ij}\,d\Omega + \int_\Gamma w_i\sigma_{ij} n_j\,d\Gamma$

Where Γ is the perimeter of Ω and n is the outward normal. This should be added (with a clearer notation). I'm not sure if this is related to the divergence theorem. —BenFrantzDale 20:22, 18 October 2005 (UTC)

Something like this definitely must be in the article. I don't recall for now how this is called, but I think that the divergence theorem is a particular case of this. From what I know, the most general version of this formula is

$\int_{\Omega} \frac{\partial u}{\partial x_i} v \,dV = \int_{\partial\Omega} u v \, \mathbf{n}_i \,d S - \int_{\Omega} u \frac{\partial v}{\partial x_i} \, dV$

from where one gets for example

$\int_{\Omega} \nabla u \cdot \mathbf{v} + u \nabla\cdot \mathbf{v}\, dV = \int_{\partial\Omega} u \mathbf{v}\cdot \mathbf{n}\, dS$

which is a very useful integration by parts formula in higher dimensions, which implies for example the Ostrogradsky-Gauss theorem and probably other cases of the Stokes theorem. Oleg Alexandrov (talk) 00:59, 19 October 2005 (UTC)

I agree that especially the last formula above should be in text. It should also be noted, that the

surface term on the right commonly goes to zero when integrating over all space and u, v "behave well". This is very often used in physics calculations, and so common than some textbooks even gloss over it. gbrandt 13:33, 26 January 2006 (UTC)

[edit] Please revise the "recursive formulation" section

I suggest the author of section 4 “recursive formulation” http://en.wikipedia.org/w/index.php?title=Integration_by_parts&action=edit&section=4 of the article Integration_by_parts to revise the recursive formula that is suspected to be mistaken or notation needs to be further clarified. It seems to differ from the corresponding formula in Mathworld: http://mathworld.wolfram.com/IntegrationbyParts.html. Consider inclusion of another term on RHS, integral of product of some functions.

[edit] Tablature Method (possible to add)

Can we add the talbulture method, a simplified way of using integration by parts. It only works for a repeating function multiplied by a non repeating function. By repeating, i mean a function that doesent settle in on zero after a fininte amount of derivatives. Can we add this to the main article? or is it somewhere else on wikipedia? Swerty 21:34, 27 March 2006 (UTC)

How about creating a new tablature method article? Oleg Alexandrov (talk) 16:25, 28 March 2006 (UTC)

[edit] Failed to parse

Can someone with more time on their hands than me please fix the code? Every equation on the page currently reads "Failed to parse" in big red letters!

They're all working fine now. Michael Hardy 19:21, 19 April 2006 (UTC)