Understanding the Weierstrass Substitution: Applications and Uses

[glebovg]

Can anyone explain Weierstrass substitution?

When would one use it and why?

Examples?

 

[paulfr]

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.

 

[glebovg]

Thanks.