10. Reduction Formulae

by M. Bourne

You may have noticed in the Table of Integrals that some integrals are given in terms of a simpler integral. These require a few steps to find the final answer.

Reduction formulae are integrals involving some variable `n`, as well as the usual `x`. They are normally obtained from using integration by parts.

We use the notation `I_n` when writing reduction formulae.

Example 1

Given the reduction formula

`I_n=intsin^nx\ dx` `=-1/ncos\ x\ sin^(n-1)x+(n-1)/nI_(n-2),`

find `intsin^4x\ dx`.

Example 2

We are given that if

`I_n=inttan^nx\ dx`,

then `I_n=1/(n-1)tan^(n-1)x-I_(n-2).`

Find `inttan^3x\ dx`.

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