Integration By Parts VS U-Substitution

[Soubriquet]

The past few examples in my review book demonstrated u-substitution to integrate trig functions. The example I'm on suddenly shows integration by parts. The book doesn't explain why this method is used over u-sub.

\intsec³x dx

In what situation am I supposed to use one method over the other?

 

[Mark44]

Integration by parts is generally more complicated than ordinary substitution, so I usually try the substitutions first before going to integration by parts. In the integral you show, there aren't any obvious choices for ordinary substitutions, so IBP is called for.

 

[Soubriquet]

Thanks for the reply.
You said "substitutions." What other substitution methods are there other than u-sub?

 

[Mark44]

Well, there is trig substitution, but what I meant was that there are often different possibilities for choices for ordinary substitutions.

 

[Bohrok]

If you don't want to use integration by parts, you could use a u-substitution and partial fractions (but probably more work):
\sec^3x = \frac{1}{\cos^3x} \cdot \frac{\cos x}{\cos x} = \frac{\cos x}{\cos^4x} = \frac{\cos x}{(\cos^2x)^2} = \frac{\cos x}{(1 - \sin^2x)^2}
Let u = sin x, then partial fractions.

 

[paulfr]

Tabular Integration by Parts is quite a bit easier to do than the classic IBP.
I recommend you learn it.

 

[Unit]

paulfr, can you provide any resources or links or even an explanation for Tabular IBP? I have never heard of it!

 

[paulfr]

http://en.wikipedia.org/wiki/Integration_by_parts
Tabular method is about 2/3 way down the page.

Jaimie Escalante used it in the movie Stand & Deliver about a Calc teacher in LA.

 

[Soubriquet]

I found an interesting pdf on Tabular IBP.
http://www.maa.org/pubs/Calc_articles/ma035.pdf

Tabular IBP is pretty neat and much faster than classic IBP in some cases. Thanks paulfr.

 

[paulfr]

Here is a video presentation of a Tabular IBP
http://www.youtube.com/watch?v=Qlht...B693596C&playnext_from=PL&playnext=1&index=47

 

[Theorem.]

U-substitution is the most simple method of substitution. IF you can't do a simple U-substitution and a product is involved, then you want to look at alternative methods, such as integration by parts.