Reduction formulae
Objective. Derive and use recurrence relations for integrals.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
A reduction formula expresses an integral Iₙ, depending on an integer n, in terms of a related integral with a smaller index such as I_(n−1) or I_(n−2), typically derived using integration by parts. Once established, the formula lets a high-index integral be evaluated by repeated application down to a base case that can be computed directly, such as I₀ or I₁. For example, for Iₙ = ∫xⁿeˣdx, integration by parts gives Iₙ = xⁿeˣ − nI_(n−1), reducing the power of x by one each time.
Worked example
Given the reduction formula Iₙ = ∫₀^1 xⁿe^x dx satisfies Iₙ = e − nI_(n−1), and I₀ = e−1, find I₂.
- 1First find I₁ using the formula with n=1: I₁ = e − 1×I₀ = e − (e−1) = 1
- 2Now find I₂ using the formula with n=2: I₂ = e − 2×I₁
- 3Substitute I₁=1: I₂ = e − 2(1)
- 4Simplify: I₂ = e − 2
Answer: I₂ = e − 2
Concept mastery
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Practice
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Start practisingCommon mistake. Students apply the reduction formula but forget to also evaluate the base case (n=0 or n=1) explicitly, leaving the final answer incomplete. Always compute down to and including the base case before combining results.
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