Deriving and using reduction formulae

Deriving and using reduction formulae

Understanding Reduction Formulae

  • Reduction formulae are an important tool in calculus, utilised to simplify the integration process of certain functions.
  • Essentially, a reduction formula allows for the reduction of the power of the function within the integral, making the integral easier to solve.
  • Named after French mathematician Pierre-Simon Laplace, they are also known as “Laplace’s Reduction Formulas”.

Derivation of Reduction Formulas

  • Reduction formulae are derived using integration by parts, which is itself a product rule for integration.
  • The basic form of integration by parts states: ∫udv = uv - ∫vdu.
  • An initial integral is turned into the sum of a simpler integral and a product.
  • The key step in deriving a reduction formula for, say, ∫xn sin(ax) dx or ∫xn e^(ax) dx, is to make a clever choice of the functions ‘u’ and ‘dv’ to substitute into the integration by parts formula.
  • The result is a formula that expresses the original integral in terms of a lower-powered integral and known functions, which is easier to evaluate.

Application of Reduction Formulae

  • Once derived, a reduction formula can greatly simplify the task of integrating any function within its scope.
  • Each application of the formula reduces the power by one, hence the term ‘reduction’.
  • Keep applying the formula until the integral becomes simple enough to be computed directly.
  • Caution! Though the reduction process simplifies, one must be alert to keep track of all terms, as it could become a lengthy process.

Examples of Reduction Formulae

  • Here are a few standard reduction formulae in integral calculus:
    1. The reduction formula for ∫xp e^(ax) dx
    2. The reduction formula for ∫x^m (ln x)^n dx
    3. The reduction formula for ∫sin^n(a) dx

Through consistent practice in using different examples and tackling numerous practice problems, your understanding of reduction formulae will enhance dramatically. An understanding of reduction formulae is vital not just for integral calculus but also for many areas of mathematics, science, and engineering where integration becomes necessary.