Understanding the Weierstrass Substitution: Applications and Uses

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SUMMARY

The Weierstrass substitution, also known as the Tangent Half Angle Method, is a technique used to convert rational trigonometric integrands into rational algebraic integrands, facilitating the integration process. This method allows the application of various standard techniques for finding antiderivatives, such as Long Division, Completing the Square, Partial Fraction Decomposition, and Trigonometric Substitution. Its effectiveness lies in simplifying complex integrals, making it a valuable tool for advanced calculus. While not commonly taught at the high school level due to time constraints, it is recognized as a powerful integration technique.

PREREQUISITES
  • Understanding of integration techniques in calculus
  • Familiarity with trigonometric functions and identities
  • Knowledge of algebraic manipulation methods
  • Basic proficiency in using substitution methods for integrals
NEXT STEPS
  • Study the derivation and application of the Weierstrass substitution in detail
  • Explore examples of integrals solved using the Tangent Half Angle Method
  • Learn about Long Division and Partial Fraction Decomposition techniques in integration
  • Investigate other substitution methods in calculus, such as Trigonometric Substitution
USEFUL FOR

Mathematics students, calculus instructors, and anyone interested in advanced integration techniques will benefit from this discussion on the Weierstrass substitution.

glebovg
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Can anyone explain Weierstrass substitution?

When would one use it and why?

Examples?
 
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Weierstrass Substitution is also referred to as the Tangent Half Angle Method.
Basically it takes a rational trigonometric integrand and converts it to a rational algebraic integrand via substitutions. The reason it is so powerful is that with Algebraic integrands you have numerous standard techniques for finding the AntiDerivative; Long Division, Complete the Square, Partial Fraction Decomposition, and Trig Substitution.

It looks harder than it really is. Just make the substitutions and watch the fallout simplify the problem. Look at Wiki here ...
http://en.wikipedia.org/wiki/Weierstrass_substitution

See the third example involving Secant and check the section "derivation" to see what dx is and why.

I do not teach this to my HS students as we do not have enough time given all the other techniques there are to learn. But this is a cool and powerful tool for Integration.
 
Thanks.
 

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