# Further Calculus: Reduction formulae

## Understanding Reduction Formulae

• In Further Calculus, the reduction formulae method is used to simplify the integration of functions, especially when dealing with functions of the form ∫xn sin(x), ∫xn cos(x), ∫xn e^x, etc.
• It involves an integral set up in a recursion formula that reduces the power of the variable by one with each iteration.
• A reduction formula can simplify calculation where a function is to be integrated multiple times, or where an unknown function is represented as an infinite series.

## Building a Reduction Formula

• A reduction formula is typically developed using integration by parts, an integral version of the product rule in differentiation.
• The formulas are generally in the form: ∫ xn f(x) dx = F(x)xn - n ∫ xn-1 F(x) dx.
• It’s important to note that developing a reduction formula involves establishing a pattern wherein the integral on the right-hand side is simpler than the one on the left.

## Applying Reduction Formulae

• Identify the right situation to apply the reduction formula. The expression must be integrable and should have a term with variable power that reduces on applying the formula.
• Carry out integration by parts on the original integral, then rewrite the result so the same integral is expressed on both sides, but the one on the right-hand side is easier.
• Repeat the process until the integral simplifies enough to be solved directly.

## Examples of Reduction Formulae

• Examples of reduction formulae include:
• The sine integral reduction formula: ∫ xn sin(x) dx = -xn cos(x) + n ∫ xn-1 cos(x) dx.
• The exponential integral reduction formula: ∫ xn e^x dx = xn e^x - n ∫ xn-1 e^x dx.

Note: The solution to an integral using reduction formulae might involve a simpler integral rather than a simple function. In such cases, evaluate the simpler integral normally to get the final answer.

## Reduction Formulae in Problem Solving

• Reduction formulae are particularly useful in solving problems with repeated integrals or series. They can help in simplifying otherwise difficult or impossible integrals.
• However, keep in mind that while reduction formulae might simplify the integrals on the surface, they often require more computation beneath the surface. As a result, they are not always the quickest or most efficient solution.