How do you know when it's time for Trignometric Subsitution ?

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Discussion Overview

The discussion revolves around identifying when to use trigonometric substitution as an integration technique. Participants explore various scenarios and conditions under which this method may simplify integrals, focusing on its application in calculus.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants suggest that trigonometric substitution should always work and can make integration easier.
  • Others propose that it is particularly useful when the integrand resembles forms like \(\frac{1}{\sqrt{1+x^2}}\) or \(\frac{1}{\sqrt{1-x^2}}\), indicating a connection to inverse trigonometric functions.
  • A participant mentions that seeing a combination of 1 and \(x^2\) often signals the potential for trigonometric or hyperbolic substitution.
  • One participant advises that if unsure about using trigonometric substitution, it may be wise to attempt solving the integral without it first, suggesting that the substitution can complicate the process if not beneficial.

Areas of Agreement / Disagreement

Participants express varying opinions on the conditions for using trigonometric substitution, with no consensus on a definitive guideline. Multiple viewpoints on its applicability and effectiveness remain present.

Contextual Notes

Some assumptions about the forms of integrals suitable for trigonometric substitution are not explicitly defined, and the discussion does not resolve the effectiveness of the technique in all cases.

jkh4
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How do you know when it's time for "Trignometric Subsitution"?

How do you know when it's time for "Trignometric Subsitution"?
 
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You should be more specific about that question.
 
it an integration technique. it should always "work". sometimes it just makes things easier.
 
jkh4 said:
How do you know when it's time for "Trignometric Subsitution"?
Whenever a trigonometric substitution simplifies your integrand significantly.
 
Usually when you see something like \frac{1}{\sqrt{1+x^2}} or \frac{1}{\sqrt{1-x^2}} or anything similar then you can think about using a trig sub. I usually think of trig sub when I see somethign that looks like the derivative of an inverse trig functions. (you should memorize those). But there are definitely many cases where a trig sub could be used...
 
basically whenever you see a 1 and an x2, you can do some sort of trig or hyperbolic trig substitution to make things easier. Of course, if you're not sure, try solving it without the substitution, and only if you get stuck should you go for it (since if it's not useful, it can be a real pain in the ass trying to figure that out).
 

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