reduction formula

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Let In be some quantity that is dependent upon the integer n(≥0). It may be possible to establish some general formula, expressing In in terms of some of the quantities In−1, In−2,…. Such a formula is a reduction formula and can be used to evaluate In for a particular value of n. The method is useful in integration. For example, if

In = 0π/2 sinnxdx,

it can be shown, by integration by parts, that In=(n−1)/n)In−2 (n≥2). It is easy to see that I0=π/2, and then the reduction formula can be used, for example, to find that

Subjects: Mathematics.

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