Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.3.5

Chapter 8, Problem 8.3.5

5. What is a reduction formula?

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A reduction formula is a mathematical expression used to simplify the computation of integrals by reducing them to simpler forms. It is often expressed as a recursive relationship between an integral involving a parameter and a simpler integral with a smaller parameter.

Reduction formulas are particularly useful for integrals involving powers, trigonometric functions, or factorials, where direct computation can be challenging.

To derive a reduction formula, start by applying integration techniques such as integration by parts, substitution, or trigonometric identities to the given integral.

Once the integral is expressed in terms of a simpler integral (usually with a reduced parameter), identify the recursive relationship and write it as a formula.

Reduction formulas are commonly used in problems requiring repeated integration, such as evaluating definite integrals or solving differential equations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Reduction Formula

A reduction formula is a recursive relationship that expresses a function in terms of its values at smaller arguments. It is particularly useful in calculus for simplifying the evaluation of integrals or sums by breaking them down into more manageable parts. By applying the reduction formula iteratively, one can often reduce complex problems to simpler ones that are easier to solve.

Integration Techniques

Integration techniques are methods used to compute integrals, which can include substitution, integration by parts, and partial fractions. Understanding these techniques is essential for applying reduction formulas effectively, as they often rely on manipulating integrals into forms that can be simplified or solved using known results. Mastery of these techniques allows for a more efficient approach to solving complex integrals.

Recursion in Mathematics

Recursion in mathematics refers to defining a function in terms of itself, allowing for the solution of problems by breaking them down into smaller, similar problems. This concept is foundational in many areas of mathematics, including calculus, where reduction formulas often utilize recursive relationships to simplify calculations. Recognizing how to apply recursion can lead to elegant solutions for otherwise complicated problems.

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Related Practice

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

48. ∫ √(9 - 4x²) dx

Textbook Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

20. ∫ sin⁻¹(x) dx

Textbook Question

7–64. Integration review Evaluate the following integrals.

20. ∫ eˣ (1 + eˣ)⁹ (1 - eˣ) dx

Textbook Question

50-53. Reduction Formulas Use integration by parts to derive the following reduction formulas:

53. ∫ lnⁿ(x) dx = x lnⁿ(x) - n ∫ lnⁿ⁻¹(x) dx

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

24. ∫ dt / √(1 + 4eᵗ)

Textbook Question

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

54. ∫ y⁴/(1 + y²) dy