A-LevelMathematicsIntegration9709/23 · Oct/Nov 2024 · Q37 Marks
The function f is defined by f(x) = tan²(½x) for 0 ≤ x < π.
- (a)Find the exact value of f'(⅓π).[3]
- (b)Find the exact value of ∫₀^(π) (f(x) + sinx)dx.[4]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.
- [3] Differentiate to obtain form k tan ½x sec² ½x M1 OE. May use identities before differentiation.
- Obtain correct tan ½x sec² ½x A1 OE. Allow unsimplified.
- Substitute π/3 to obtain 4√3 A1
- [4] Express integrand as sec²½x-1+sinx B1
- Integrate to obtain k₁tan ½x - x + k₂cosx M1 Where k₁k₂ ≠ 0.
- Obtain correct 2tan ½x - x - cos x A1
- Apply limits correctly to obtain 3-½π or exact equivalent A1
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