To determine The value of integral . Explanation Given information: ∫ 0 π 8 sin 4 x d x 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 ) d x = − cos ( x ) sin n − 1 ( x ) n + n − 1 n ∫ sin n − 2 ( x ) d x sin 2 ( x ) = 1 − cos ( 2 x ) 2 Calculation: Express the integral as below. ∫ 0 π 8 sin 4 x d x = ∫ 0 π 8 sin 4 x d x = 8 ∫ 0 π sin 4 x d x ∫ 0 π 8 sin 4 x d x = 8 ( − [ cos ( x ) sin 3 ( x ) 4 ] 0 π + 3 4 ∫ 0 π sin 2 ( x ) d x ) (1) Consider the second part of the integral from the equation (1) and express it as below. ∫ 0 π sin 2 ( x ) d x = ∫ 0 π 1 − cos ( 2 x ) 2 d x = 1 2 ∫ 0 π 1 − cos ( 2 x ) d x ∫ 0 π sin 2 ( x ) d x = 1 2 ( ∫ 0 π d x − ∫ 0 π cos ( 2 x ) d x ) (2) Consider the second part of the integral from the equation (2) and express it as below. ∫ 0 π cos ( 2 x ) d x (3) Assume the variable u as u = 2 x Differentiate the above equation with respect to x

Thomas' Calculus: Early Transcendentals Plus Mymathlab With Pearson Etext -- Access Card Package (14th Edition) (hass, Heil & Weir, Thomas' Calculus Series)

14th Edition

ISBN: 9780134665573

Author: Joel R. Hass, Christopher D. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 8.3, Problem 17E

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The value of integral.

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