A-LevelFurther MathematicsIntegration9231/12 · May/June 2016 · Q59 Marks
Let In = ∫ (from 0 to (π/2)) cosⁿ x sin²x dx, for n ≥ 0. By differentiating cosⁿ⁻¹ x sin³ x with respect to x, prove that (n + 2)In = (n − 1)In−2 for n ≥ 2. Hence find the exact value of I₄.
- Let In = ∫ (from 0 to (π/2)) cosⁿ x sin²x dx, for n ≥ 0. By differentiating cosⁿ⁻¹ x sin³ x with respect to x, prove that (n + 2)In = (n − 1)In−2 for n ≥ 2.[5]
- Hence find the exact value of I₄.[4]
Mark scheme
Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.
- [M1A1] d/dx (cosⁿ⁻¹ x sin³ x) = -(n-1)cosⁿ⁻²xsin⁴x + 3cosⁿ xsin²x
- [M1] [cosⁿ⁻¹xsin³x] (from 0 to π/2) = -(n-1)∫cosⁿ⁻²xsin²x(1-cos²x)dx +3Iₙ
- [M1] ⇒0=-(n-1) Iₙ₋₂ + (n-1)Iₙ + 3Iₙ
- [A1] ⇒(n+2)Iₙ =(n-1) Iₙ₋₂ (AG)
- [M1] I₀ = ∫ sin²x dx = ½ ∫ (1-cos 2x)dx
- [A1] = [x/2 - sin(2x)/4] (from 0 to π/2) = π/4
- [M1A1] I₂ = ¼ * I₀ = π/16; I₄ = ½ * I₂ = π/32
Write your answer and get it marked
On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.
Sign up free →