A-LevelMathematicsAlgebra9709/33 · Oct/Nov 2022 · Q1110 Marks
Let f(x) = (5 - x + 6x²)/((3 - x)(1 + 3x²)).
- (a)Express f(x) in partial fractions.[5]
- (b)Find the exact value of ∫(from 0 to 1) f(x) dx, simplifying your answer.[5]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.
- B1 State or imply the form A/(3-x) + (Bx+C)/(1+3x²)
- M1 Use a correct method to find a constant
- A1 Obtain one of A = 2, B = 0 and C = 1
- A1 Obtain a second value
- A1 Obtain the third value
- B1 FT Integrate and obtain term -2ln(3-x)
- M1 Obtain term of the form b tan⁻¹(√3x)
- A1 FT Obtain term (1/√3)tan⁻¹(√3x)
- M1 Substitute limits correctly in an integral with terms a ln(3-x) and b tan⁻¹(√3x), where ab ≠ 0
- A1 Obtain answer 2ln(3/2) + (1/(3√3))π, or equivalent
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