Explanation Given information: ∫ − π / 2 π / 2 cos x cos 7 x d x Formula: ∫ x a d x = x a + 1 a + 1 + C ∫ sin x d x = − cos x + C cos m x cos n x = 1 2 [ cos ( m − n ) x + cos ( m + n ) x ] Calculation: Express the integral as below. ∫ − π / 2 π / 2 cos x cos 7 x d x = ∫ − π / 2 π / 2 1 2 ( cos ( 1 − 7 ) x + cos ( 1 + 7 ) x ) d x = ∫ − π / 2 π / 2 1 2 ( cos ( − 6 x ) + cos 8 x ) d x ∫ − π / 2 π / 2 cos x cos 7 x d x = 1 2 ∫ − π / 2 π / 2 cos 6 x d x + 1 2 ∫ − π / 2 π / 2 cos 8 x d x (1) Consider the first part of integral in equation (1) as below. 1 2 ∫ − π / 2 π / 2 cos 6 x d x (2) Assume the variable u as 6 x . u = 6 x Differentiate the above equation with respect to x . d u d x = 6 d x = d u 6 When the value of x is π 2 , the value of u is 3 π . When the value of x is − π 2 , the value of u is − 3 π . Substitute d u 6 for d x and u for 6 x by changing the limits in the equation (2)

Thomas' Calculus Format: Unbound (saleable) With Access Card

14th Edition

ISBN: 9780134768762

Author: Hass, Joel R.^heil, Christopher D.^weir, Maurice D.

Publisher: Prentice Hall

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Chapter 8.3, Problem 56E

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