Integration By Parts VS U-Substitution

In summary, the conversation revolves around different methods of integration, specifically u-substitution and integration by parts. The speaker is struggling to understand when to use one method over the other, and the other person suggests trying ordinary substitutions first before moving on to integration by parts. They also mention trig substitution and partial fractions as other possibilities for substitutions. The concept of Tabular Integration by Parts is introduced as a faster alternative to classic IBP, with resources and a demonstration provided. The conversation concludes with a recommendation to learn Tabular IBP.
  • #1
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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.

[tex]\int[/tex]sec3x dx

In what situation am I supposed to use one method over the other?
 
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  • #2
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
Thanks for the reply.
You said "substitutions." What other substitution methods are there other than u-sub?
 
  • #4
Well, there is trig substitution, but what I meant was that there are often different possibilities for choices for ordinary substitutions.
 
  • #5
If you don't want to use integration by parts, you could use a u-substitution and partial fractions (but probably more work):
[tex]\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}[/tex]
Let u = sin x, then partial fractions.
 
  • #6
Tabular Integration by Parts is quite a bit easier to do than the classic IBP.
I recommend you learn it.
 
  • #7
paulfr, can you provide any resources or links or even an explanation for Tabular IBP? I have never heard of it!
 
  • #9
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.
 
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  • #11
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.
 

1. What is the difference between Integration By Parts and U-Substitution?

Integration By Parts and U-Substitution are two different techniques used to solve integrals. Integration By Parts involves splitting a complicated integral into two simpler integrals, while U-Substitution involves substituting a variable to simplify the integral.

2. When should I use Integration By Parts instead of U-Substitution?

You should use Integration By Parts when the integral involves a product of two functions, and the derivative of one of the functions becomes simpler after applying the product rule. In other words, if one function gets easier to integrate and the other function gets easier to differentiate, Integration By Parts should be used.

3. Is Integration By Parts more difficult than U-Substitution?

It depends on the integral. Some integrals may be easier to solve using Integration By Parts, while others may be easier using U-Substitution. It is important to practice and become familiar with both techniques in order to determine which one to use for a particular integral.

4. Can I use both Integration By Parts and U-Substitution in the same integral?

Yes, it is possible to use both techniques in the same integral. This is known as the "double substitution" method, where you use U-Substitution to simplify the integral and then use Integration By Parts to solve the remaining integral.

5. Are there any shortcuts or tips for using Integration By Parts and U-Substitution?

One tip for Integration By Parts is to use the acronym "LIATE" (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential) to determine which function to differentiate and which function to integrate. For U-Substitution, a common tip is to choose a substitution that will result in the integral being in the form of a known derivative. It is also helpful to practice and become familiar with the common substitutions used in U-Substitution.

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