Question

Metode parsial
1. ∫ (x + 3) cos(2x - π)
2. ∫ (x² + 1) cosx dx

Answer 1

Verified answer

Bentuk integral Parsial
∫ u dv = u v - ∫ v du

1) ∫ (x + 3) cos (2x - π) dx

u = x + 3 ............. dv = cos (2x - π) dx
du = dx ................. v = 1/2 sin (2x - π)

= u v - ∫ v du
= (x + 3) . 1/2 sin (2x - π) - ∫ 1/2 sin (2x - π) dx
= 1/2 (x + 3) sin (2x - π) - (-1/4 cos (2x - π)) + C
= 1/2 (x + 3) sin (2x - π) + 1/4 cos (2x - π) + C
atau
= 1/2 x sin (2x - π) + 3/2 sin (2x - π) + 1/4 cos (2x - π) + C

2) ∫ (x² + 1) cos x dx

u = x² + 1 ......... dv = cos x dx
du = 2x dx ...... v = sin x

= u . v - ∫ v du
= (x² + 1) . sin x - ∫ sin x . 2x dx
= (x² + 1) sin x - ∫ 2x sin x dx

u = 2x ........... dv = sin x dx
du = 2 dx ...... v = - cos x

= (x² + 1) sin x - [u v - ∫ v du]
= (x² + 1) sin x - [2x . - cos x - ∫ (- cos x) 2 dx]
= (x² + 1) sin x + 2x cos x + ∫ 2 cos x dx]
= (x² + 1) sin x + 2x cos x + 2 sin x + C
atau
= x² sin x + sin x + 2x cos x + 2 sin x + C
= x² sin x + 2x cos x + 3 sin x + C

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