Explanation Given information: ∫ 0 π sin 5 x 2 d x Formula: ∫ cos x d x = sin x + C ∫ sin x d x = − cos x + C ∫ x a d x = x a + 1 a + 1 sin 2 ( x ) = 1 − cos 2 ( x ) Calculation: Evaluate the integral. ∫ 0 π sin 5 x 2 d x (1) Assume the variable u as u = x 2 Differentiate the above equation. d u d x = ( 1 ) 2 d u = d x 2 d x = 2 d u When the value of x is 0, the value of u is 0. When the value of x is π , the value of u is π 2 . Substitute 2 d u for d x and u for x 2 and change the limits in the equation (1). ∫ 0 π sin 5 x 2 d x = ∫ 0 π 2 sin 5 ( u ) ( 2 d u ) = 2 ∫ 0 π 2 sin 5 ( u ) d u = 2 ∫ 0 π 2 sin 4 ( u ) sin ( u ) d u = 2 ∫ 0 π 2 ( sin 2 ( u ) ) 2 sin ( u ) d u ∫ 0 π sin 5 x 2 d x = 2 ∫ 0 π 2 ( sin 2 ( u ) ) 2 sin ( u ) d u ∫ 0 π sin 5 x 2 d x = 2 ∫ 0 π 2 ( 1 − cos 2 ( u ) ) 2 sin ( u ) d u (2) Assume the variable v as v = cos u Differentiate the above equation

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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