Explanation Formulas used: Trigonometric identities for sine and cosine functions : cos A cos B = 1 2 ( cos ( A + B ) + cos ( A − B ) ) sin A cos B = 1 2 ( sin ( A + B ) + sin ( A − B ) ) cos 2 x = 1 2 ( 1 + cos 2 x ) cos x sin x = 1 2 sin 2 x Calculation: Consider the given integrals ∫ − L L cos ( m π x L ) cos ( n π x L ) d x and ∫ − L L cos ( m π x L ) sin ( n π x L ) d x . Compute the integral ∫ − L L cos ( m π x L ) cos ( n π x L ) d x as shown below. Case 1: m ≠ n . ∫ − L L cos ( m π x L ) cos ( n π x L ) d x = ∫ − L L 1 2 ( cos ( ( m + n ) π x L ) + cos ( ( m − n ) π x L ) ) d x = 1 2 ( ∫ − L L cos ( ( m + n ) π x L ) d x + ∫ − L L cos ( ( m − n ) π x L ) d x ) = 1 2 ( ( sin ( ( m + n ) π x L ) ( m + n ) π L ) + ( sin ( ( m − n ) π x L ) ( m − n ) π L ) ) − L L = L 2 π ( sin ( ( m + n ) π x L ) m + n + sin ( ( m − n ) π x L ) m − n ) − L L Apply the limits and evaluate the integral ∫ − L L cos ( m π x L ) cos ( n π x L ) d x as shown below. ∫ − L L cos ( m π x L ) cos ( n π x L ) d x = L 2 π ( ( sin ( ( m + n ) π L L ) m + n + sin ( ( m − n ) π L L ) m − n ) − ( sin ( ( m + n ) π ( − L ) L ) m + n + sin ( ( m − n ) π ( − L ) L ) m − n ) ) = L 2 π ( ( sin ( m + n ) π m + n + sin ( m − n ) π m − n ) − ( sin ( − ( m + n ) π ) m + n + sin ( − ( m − n ) π ) m − n ) ) = L 2 π ( ( 0 + 0 ) − ( 0 + 0 ) ) = 0 Case 2: m = n . In this case the integral ∫ − L L cos ( m π x L ) cos ( n π x L ) d x becomes ∫ − L L cos 2 ( m π x L ) d x . Compute the integral ∫ − L L cos 2 ( m π x L ) d x as shown below. ∫ − L L cos 2 ( m π x L ) d x = ∫ − L L 1 2 ( 1 + cos 2 ( m π x L ) ) d x = 1 2 ( ∫ − L L 1 d x + ∫ − L L cos 2 ( m π x L ) d x ) = 1 2 ( ( x ) + sin 2 ( m π x L ) 2 m π L ) − L L Apply the limits and evaluate the integral ∫ − L L cos 2 ( m π x L ) d x as shown below. ∫ − L L cos 2 ( m π x L ) d x = 1 2 ( ( L ) + sin 2 ( m π L L ) 2 m π L − ( − L ) + sin 2 ( m π ( − L ) L ) 2 m π L ) = 1 2 ( L + L sin ( 2 m π ) 2 m π + L + L sin ( − 2 m π ) 2 m π ) = 1 2 ( L + 0 + L + 0 ) = 1 2 ( 2 L ) = L Therefore, the value of the integral ∫ − L L cos ( m π x L ) cos ( n π x L ) d x = { 0 , m ≠ n L , m = n }

Elementary Differential Equations and Boundary Value Problems, Enhanced

11th Edition

ISBN: 9781119381648

Author: Boyce

Publisher: WILEY

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Chapter 10.2, Problem 12P

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To verify: The equations ∫−LLcos(mπxL)cos(nπxL)dx={0, m≠nL, m=n} and ∫−LLcos(mπxL)sin(nπxL)dx=0 for all m, n.

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To verify: The equations ∫ − L L cos ( m π x L ) cos ( n π x L ) d x = { 0 , m ≠ n L , m = n } and ∫ − L L cos ( m π x L ) sin ( n π x L ) d x = 0 for all m , n .

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To verify: The equations ∫−LLcos(mπxL)cos(nπxL)dx={0, m≠nL, m=n} and ∫−LLcos(mπxL)sin(nπxL)dx=0 for all m, n.

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