Reduction Formulae

A reduction formula is a recurrence relation that expresses an integral I_n (involving a parameter n) in terms of I_(n-1), I_(n-2), or similar. This powerful technique allows you to evaluate families of integrals systematically without integrating from scratch each time.

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1 What Is a Reduction Formula?

A reduction formula takes the form:

I_n = (some expression involving n) + (some coefficient) * I_(n-k)

where I_n = integral of some expression involving a parameter n, and k is usually 1 or 2.

By applying the formula repeatedly, you reduce the index n down to a base case (I_0 or I_1) that is easy to evaluate.

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2 Deriving a Reduction Formula

The standard approach is:

1. Write down I_n = integral of the given expression.
2. Apply integration by parts (or another technique) to express I_n in terms of I_(n-1) or I_(n-2).
3. State the reduction formula clearly.
4. Find the base case(s).
5. Use the formula to compute specific values.

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3 Worked Example 1 — I_n = integral from 0 to 1 of x^n e^x dx

Step 1: Set up the integral.