To determine The value of integral. Explanation Given information: ∫ 0 π 8 sin 4 y cos 2 y d y Formula: ∫ a b cos x d x = [ sin x ] a b ∫ a b x n d x = [ x n + 1 n + 1 ] a b ∫ sin n ( x ) cos 2 ( x ) d x = − sin n − 1 ( x ) cos 3 ( x ) n + 2 + n − 1 n + 2 ∫ sin n − 2 ( x ) cos 2 ( x ) d x cos 2 ( x ) sin 2 ( x ) = 1 − cos ( 4 x ) 8 Calculation: Express the integral as below. ∫ 0 π 8 sin 4 y cos 2 y d y = 8 ( − [ sin 3 ( y ) cos 3 ( y ) 6 ] 0 π + 3 6 ∫ 0 π sin 2 ( y ) cos 2 ( y ) d y ) (1) Consider second part of the integral term from the equation (1). ∫ 0 π sin 2 ( y ) cos 2 ( y ) d y (2) Rewrite the above equation as below. ∫ 0 π sin 2 ( y ) cos 2 ( y ) d y = ∫ 0 π 1 − cos ( 4 y ) 8 d y = 1 8 ∫ 0 π 1 − cos ( 4 y ) d y ∫ 0 π sin 2 ( y ) cos 2 ( y ) d y = 1 8 ( ∫ 0 π d y − ∫ 0 π cos ( 4 y ) d y ) (3) Consider second part of the integral term from the equation (3). ∫ 0 π cos ( 4 y ) d y (4). Assume the variable u as u = 4 y . Differentiate the above equation with respect to y . d u d y = 4 d y = d u 4 When the value of y is 0, the value of u is 0

Thomas' Calculus - MyMathLab Integrated Review

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ISBN: 9780134786223

Author: Hass

Publisher: PEARSON

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Chapter 8.3, Problem 20E

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