Reduction formulae

Reduction Formulae Basics

  • Reduction formulae are typically used when dealing with power functions or functions of the form ax^n.
  • These formulae allow to express the integral of a function in terms of an integral of lower power.
  • Generally, the aim is to simplify a complicated integral into a less complicated integral.

Creating a Reduction Formula

  • A reduction formula can be found through integration by parts; a method derived from product rule for differentiation.
  • The general formula is ∫ u dv = uv - ∫ v du where u and v are functions, and du and dv denote their differentials.
  • Choose u and dv wisely, such that their derivative and antiderivative respectively are simpler than the original and can be integrated or differentiated easily.

Different Types of Reduction Formulae

  • The reduction formula for ∫ sin^n(x) dx, where n is a positive integer, is obtained by expressing sin^n(x) as sin^(n-1)(x) * sin(x) and then using the reduction formula.
  • Another common reduction formula involves the integral of x^n * e^ax dx, where a can be any real number and n is a positive integer.
  • There are also reduction formulae for more general power functions such as ∫ x^n * f(x) dx and sequences such as ∫ cos^n(ax) dx.

Using Reduction Formulae

  • Remember to write out the integral you’re trying to solve at each step of the process.
  • Be careful with negative signs. The reduction formula often involves subtraction, which can flip the sign of the integral you’re using to reduce your original integral.
  • Substituting back after using your reduction formula correctly is a key step and sometimes this step could also lead to another integral which may require another use of the reduction formula.

Applications of Reduction Formulae

  • Reduction formulae are commonly used in physics and engineering when solving differential equations that involve repetitive integrals.
  • They are widely used in mathematics for calculations related to probability and statistics.
  • By saving computational time and reducing complexity, they aid computational efficiency in the field of numerical methods.