Understanding the Weierstrass Substitution: Applications and Uses

In summary, Weierstrass Substitution, also known as the Tangent Half Angle Method, is a technique used to convert a rational trigonometric integrand into a rational algebraic integrand through substitutions. This makes it easier to find the AntiDerivative using standard techniques such as Long Division, Complete the Square, Partial Fraction Decomposition, and Trig Substitution. It may seem difficult at first, but by making the necessary substitutions, the problem can be simplified. The third example on the Wikipedia page shows the use of Secant and explains the "derivation" of dx and its purpose. This technique is not typically taught to high school students due to time constraints, but it is a useful and powerful tool for Integration.
  • #1
glebovg
164
1
Can anyone explain Weierstrass substitution?

When would one use it and why?

Examples?
 
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  • #2
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.
 
  • #3
Thanks.
 

1. What is the Weierstrass substitution?

The Weierstrass substitution is a mathematical technique used to transform a complicated expression into a simpler one by substituting a new variable for the existing one. It is commonly used in integration problems involving trigonometric functions.

2. How does the Weierstrass substitution work?

The Weierstrass substitution works by replacing the existing variable with a new variable in such a way that it simplifies the expression. The new variable is usually chosen in a way that eliminates the trigonometric functions, making the integral more manageable.

3. When should the Weierstrass substitution be used?

The Weierstrass substitution should be used when solving integrals involving trigonometric functions or expressions that can be simplified by using trigonometric identities. It can also be used to solve differential equations or to simplify complicated expressions in general.

4. What are the advantages of using the Weierstrass substitution?

The main advantage of using the Weierstrass substitution is that it simplifies complicated expressions, making them easier to integrate or solve. It also allows for the use of other integration techniques, such as partial fractions, which may not have been possible with the original expression.

5. Are there any limitations to the Weierstrass substitution?

While the Weierstrass substitution can be a powerful tool in solving integrals and simplifying expressions, it is not always applicable. It may not work for all trigonometric functions or expressions, and it may not always lead to a simpler expression. In some cases, other integration techniques may be more suitable.

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