The Weierstrass substitution

The Weierstrass substitution

Definition

  • The Weierstrass substitution is a technique in integral calculus for simplifying complex trigonometric integrals into algebraic ones by using a particular substitution. The technique is named after German mathematician Karl Weierstrass.
  • This substitution is commonly defined as: t = tan(x/2), which allows us to express sin(x), cos(x) and dx in terms of t.

Application and Components

  • Using the substitution, we can express sin(x), cos(x) and dx as: sin(x)=2t/(1+t²), cos(x)=(1-t²)/(1+t²) and dx=2dt/(1+t²) respectively. Note that these expressions are completely in terms of t, eliminating any x variable from the integral, thus simplifying the calculation.
  • This substitution is useful whenever the integrand involves complicated functions containing sin(x), cos(x), or tan(x).

Effect on Integral Bounds

  • When the substitution is applied, the bounds of the integral may change, and in such cases, it is crucial to ensure the correct adjustment. For example, if the original bounds were given in terms of x, they should be converted to values of t by using the defined substitution.

Solving the Integral

  • After substitution, the integral simplifies to an algebraic form which can be solved using standard integration techniques. The method usually makes it easier to solve the integral as it removes the complications brought by trigonometric functions.
  • After solving, it is important to substitute back the original variable to express the solution in terms of x.

Note on Complexity

  • It’s worth mentioning that although the Weierstrass substitution is a powerful technique, it sometimes does not simplify the integral. Consequentially, the resulting integral might be more complex than the original. So, it is desirable to choose wisely when to apply this substitution.

Remember, the Weierstrass substitution is one of many tools to simplify integrals, and it is beneficial to familiarize yourself with different techniques to efficiently solve different types of integrals.