A-LevelFurther MathematicsIntegration9231/12 · Oct/Nov 2015 · Q912 Marks
It is given that I_n = ∫ᵉ₁ (ln x)ⁿ dx for n ≥ 0. Show that I_n = (n − 1)[I_{n-2} − I_{n-1}] for n ≥ 2. Hence find, in an exact form, the mean value of (ln x)³ with respect to x over the interval 1 ≤ x ≤ e.
- (a)Show that I_n = (n − 1)[I_{n-2} − I_{n-1}] for n ≥ 2.[6]
- (b)Hence find, in an exact form, the mean value of (ln x)³ with respect to x over the interval 1 ≤ x ≤ e.[6]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2015). The mark or mark type is shown in brackets.
- [B1] (a) ∫lnx dx = xlnx - x
- [M1] (a) Apply integration by parts with u=(lnx)ⁿ⁻¹ and dv=lnx dx
- [A1] (a) Correctly evaluate boundary terms (resulting in 0)
- [M1] (a) Simplify the integral term using (lnx - 1)
- [A1] (a) Separate the integral into I_{n-1} and I_{n-2}
- [A1] (a) Final AG result achieved: (n-1)[I_{n-2} - I_{n-1}]
- [B1] (b) I₀ = [x]₁ᵉ = e-1
- [B1] (b) I₁ = [xlnx - x]₁ᵉ = 1
- [M1] (b) I₂ = 1(I₀ - I₁) = e - 1 - 1 = e - 2
- [A1] (b) I₃ = 2(I₁ - I₂) = 2(1 - (e - 2)) = 6 - 2e
- [M1] (b) MV = I₃/(e-1)
- [A1] (b) = (6 - 2e)/(e-1)
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