Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.R.125a
Chapter 8, Problem 8.R.125a

125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.
a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

Verified step by step guidance
1
Step 1: Begin by recalling the reduction formula for integrals involving powers of trigonometric functions. The goal is to express the integral of sin^m(x) in terms of the integral of sin^(m−2)(x). Start with the integral I = ∫ from 0 to π of (sin^m(x)) dx.
Step 2: Use integration by parts to simplify the integral. Let u = sin^(m−1)(x) and dv = sin(x) dx. Then, compute du = (m−1)sin^(m−2)(x)cos(x) dx and v = −cos(x). Substitute these into the integration by parts formula: ∫ u dv = uv − ∫ v du.
Step 3: Evaluate the boundary terms uv at x = 0 and x = π. Note that sin(0) = sin(π) = 0, so the boundary terms vanish. This simplifies the integral to −∫ from 0 to π of (−cos(x) * (m−1)sin^(m−2)(x)cos(x)) dx.
Step 4: Simplify the remaining integral. Combine the cos^2(x) term and use the Pythagorean identity cos^2(x) = 1 − sin^2(x) to express the integral in terms of sin^(m−2)(x). This leads to a recursive relationship between the integrals of sin^m(x) and sin^(m−2)(x).
Step 5: Factor out constants and simplify the recursive formula to show that ∫ from 0 to π of (sin^m(x)) dx = (m−1)/m × ∫ from 0 to π of (sin^(m−2)(x)) dx. This completes the proof of the reduction formula.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4m
Was this helpful?

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 an integral in terms of another integral with a lower power or degree. In this context, it allows us to simplify the computation of integrals involving powers of sine by relating them to integrals of lower powers, making it easier to evaluate complex integrals step by step.
Recommended video:
Guided course
5:59
Recursive Formulas

Definite Integral

A definite integral calculates the accumulation of a quantity, represented as the area under a curve, between two specified limits. In this case, the integral from 0 to π of sin^m x provides a specific numerical value that represents the area under the curve of sin^m x over the interval [0, π], which is crucial for applying the reduction formula.
Recommended video:
05:43
Definition of the Definite Integral

Wallis Product

The Wallis product is an infinite product that relates to the values of sine and cosine functions and is used to derive various results in calculus, particularly in evaluating integrals. It is significant in the context of this problem as it connects the reduction formula to the broader implications of sine integrals and their convergence properties.
Recommended video:
05:18
The Product Rule
Related Practice
Textbook Question

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

1
views
Textbook Question

106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

74. ∫ dx/√(√(1 + √x))

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

40. ∫ (x² - 4)/(x + 4) dx

Textbook Question

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

e. The best approach to evaluating ∫(x³ + 1)/(3x²) dx is to use the change of variables u = x³ + 1.

Textbook Question

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

102. About the y-axis