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junior college 1 | H2 Maths
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Date Posted: 2 years ago
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Integration by parts
∫ ln(x) / (2x + 1)³ dx
= ∫ ln(x) (2x + 1)-³ dx
= ln(x) ∫ (2x + 1)-³ dx - ∫ [ d/dx ln(x) · ∫ (2x + 1)-³ dx ] dx
= ln(x) (-¼ (2x + 1)-²) - ∫ 1/x · (-¼(2x + 1)-²) dx
= ¼∫ 1/x(2x + 1)² dx - ln(x) / 4(2x + 1)²
Decompose the integrand using partial fractions (you should know how)
= ¼ ∫ (1/x - 2/(2x + 1) - 2/(2x + 1)² )dx - ln(x) / 4(2x + 1)²
= ¼ ( ln(x) - ln(2x + 1) + 1/(2x + 1) ) - ln(x) / 4(2x + 1)²
= ¼ ln(x) - ¼ln(2x + 1) + 1/4(2x + 1) - ln(x) / 4(2x + 1)² + c
∫ ln(x) / (2x + 1)³ dx
= ∫ ln(x) (2x + 1)-³ dx
= ln(x) ∫ (2x + 1)-³ dx - ∫ [ d/dx ln(x) · ∫ (2x + 1)-³ dx ] dx
= ln(x) (-¼ (2x + 1)-²) - ∫ 1/x · (-¼(2x + 1)-²) dx
= ¼∫ 1/x(2x + 1)² dx - ln(x) / 4(2x + 1)²
Decompose the integrand using partial fractions (you should know how)
= ¼ ∫ (1/x - 2/(2x + 1) - 2/(2x + 1)² )dx - ln(x) / 4(2x + 1)²
= ¼ ( ln(x) - ln(2x + 1) + 1/(2x + 1) ) - ln(x) / 4(2x + 1)²
= ¼ ln(x) - ¼ln(2x + 1) + 1/4(2x + 1) - ln(x) / 4(2x + 1)² + c
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