# 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.