# Integration By Parts VS U-Substitution

1. Jul 7, 2010

### 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.

$$\int$$sec3x dx

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

Last edited: Jul 7, 2010
2. Jul 8, 2010

### Staff: Mentor

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.

3. Jul 8, 2010

### Soubriquet

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

4. Jul 8, 2010

### Staff: Mentor

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

5. Jul 8, 2010

### 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.

6. Jul 8, 2010

### paulfr

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

7. Jul 8, 2010

### Unit

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

8. Jul 8, 2010

9. Jul 8, 2010

### Soubriquet

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

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

Last edited by a moderator: May 4, 2017
10. Jul 9, 2010

### paulfr

11. Jul 9, 2010

### 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.