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: \(\displaystyle \frac{4}{x+ 2}+ \frac{1}{x+ 5}+ \frac{2}{(x+ 5)^{2}}\)

(b)

Integrate the decomposed terms individually. Differentiating the result recovers the original rational function. K is arbitrary.

Answer: \(\displaystyle 4 \log{\left(x + 2 \right)} + \log{\left(x + 5 \right)} - \frac{2}{x + 5}+K\)

(c)

Use the exact endpoint values of the stated primitive; all denominator factors are nonzero on [0,1].

Answer: \(\displaystyle \frac{1}{15} + \log{\left(\frac{243}{40} \right)}\)

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.