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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 8.3.71c
Chapter 8, Problem 8.3.71c

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0

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1
Step 1: Begin by recalling the trigonometric product-to-sum identities. Specifically, the identity for the product of sine and cosine: sin(mx)cos(nx) = (1/2)[sin((m+n)x) + sin((m-n)x)]. This will help simplify the integral.
Step 2: Substitute the product-to-sum identity into the integral. The integral becomes: ∫ sin(mx)cos(nx) dx = (1/2)∫ [sin((m+n)x) + sin((m-n)x)] dx.
Step 3: Split the integral into two separate integrals: (1/2)∫ sin((m+n)x) dx + (1/2)∫ sin((m-n)x) dx. This allows us to evaluate each term independently.
Step 4: Evaluate each integral. Recall that the integral of sin(kx) over the interval [0, π] is zero when k is an integer and k ≠ 0. Since m + n and m - n are integers, both integrals evaluate to zero.
Step 5: Conclude that the original integral evaluates to zero: ∫ sin(mx)cos(nx) dx = 0. This proves the orthogonality relation for the given conditions.

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Key Concepts

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

Orthogonality of Functions

Orthogonality in the context of functions refers to the property that two functions are orthogonal if their inner product (integral of their product over a specified interval) equals zero. This concept is crucial in Fourier series, as it allows different sine and cosine functions to be treated independently, simplifying the analysis of periodic functions.
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Fourier Series

A Fourier series is a way to represent a periodic function as a sum of sine and cosine functions. The coefficients of these sine and cosine terms are determined through integrals, and the orthogonality of these functions ensures that each coefficient can be calculated independently, leading to a unique representation of the original function.
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Properties of Sine and Cosine Functions

Sine and cosine functions have specific properties, including periodicity and symmetry, which play a significant role in their orthogonality. For instance, the integral of the product of sine and cosine functions over a complete period is zero, particularly when the frequencies (m and n) are different, which is essential for proving the orthogonality relations in Fourier series.
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Related Practice
Textbook Question

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

c. ∫ v du = u·v - ∫ u dv

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Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

46. ∫(0 to 2) x⁴ dx; n = 30

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

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Textbook Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

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Textbook Question

3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?

c. A factor of (x² + 2x + 6) in the denominator

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Textbook Question

94. [Use of Tech] Skydiving A skydiver has a downward velocity given by v(t) = V_T [(1 - e^(-2gt/V_T))/(1 + e^(-2gt/V_T))],

where t = 0 is the instant the skydiver starts falling, g = 9.8 m/s² is the acceleration due to gravity, and V_T is the terminal velocity of the skydiver.

c. Verify by integration that the position function is given by

s(t) = V_T t + (V_T²/g) ln[(1 + e^(-2gt/V_T))/2],

where s'(t) = v(t) and s(0) = 0.

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Textbook Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.

63
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