Explanation Given information: ∫ x cos 3 x d x Formula: ∫ x a d x = x a + 1 a + 1 + C ∫ sin x d x = − cos x + C ∫ cos x d x = sin x + C cos 2 x = 1 − sin 2 x sin 2 x = 1 − cos 2 x d d x sin x = cos x d d x cos x = − sin x d f ( u ) d x = d f d u ⋅ d u d x ∫ u ( x ) v ′ ( x ) d x = u ( x ) v ( x ) − ∫ v ( x ) u ′ ( x ) d x (1) Calculation: Express the integral as below. ∫ x cos 3 x d x (2) Let us take the term x as u ( x ) . u ( x ) = x Differentiate the term u with respect to x . u ' ( x ) = d x Let us take the term cos 3 x as v ′ ( x ) . Integrate the term with respect to x . ∫ v ′ ( x ) = ∫ cos 3 x d x = ∫ cos 2 x cos x d x ∫ v ′ ( x ) = ∫ [ 1 − sin 2 ( x ) ] cos x d x (3) Assume the variable u as u = sin x . Differentiate the above equation with respect to x . d u d x = cos x d u = cos x d x Substitute d u for cos x d x and u for sin x in equation (3). ∫ v ′ ( x ) = ∫ ( 1 − u 2 ) d u = [ u − u 3 3 ] Substitute sin x for u . v ( x ) = sin x − 1 3 sin 3 x Substitute x for u ( x ) , cos 3 x for v ′ ( x ) d x for u ′ ( x ) and sin x − 1 3 sin 3 x for v ( x ) in equation (1). ∫ x cos 3 x d x = x sin x − 1 3 sin 3 x − ∫ [ sin x − 1 3 sin 3 x ] d x + C (4) Consider the second part of the above integral as below

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Author: Hass

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