Hint
Use the complete partial-fraction form. Multiply by the complete denominator and compare coefficients or substitute a pole.
Worked solution
(a)
Multiply the proposed decomposition by the denominator, equate coefficients and solve for A, B and C; substitution back gives the original numerator.
Answer: \(\displaystyle \frac{3}{x+ 1}+ \frac{5}{x+ 4}+ \frac{1}{x+ 6}\)
(b)
Integrate the decomposed terms individually. Differentiating the result recovers the original rational function. K is arbitrary.
Answer: \(3 \log{\left(x + 1 \right)} + 5 \log{\left(x + 4 \right)} + \log{\left(x + 6 \right)}+K\)
(c)
Use the exact endpoint values of the stated primitive; all denominator factors are nonzero on [0,1].
Answer: \(\displaystyle \log{\left(\frac{21875}{768} \right)}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Choose a complete decomposition and solve for every coefficient.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Repeated factors require all lower powers.
(b) · 2 marks
- 1 mark: Integrate every decomposed term with its scale and include an arbitrary constant.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A quadratic numerator may need to split into a logarithmic derivative and an inverse tangent term.
(c) · 2 marks
- 1 mark: Use a valid primitive at both endpoints, with no pole in the interval.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Repeated factors require all lower powers.
Total: 6 marks.