A2 June 2022 Paper 1 Q6

EdexcelCurrent spec7 marksIntegration

(a) Express as partial fractions\[\frac{2x^2 + 3x + 6}{(x + 1)\left(x^2 + 4\right)}\] (3)

(b) Hence, show that\[\int_0^2 \frac{2x^2 + 3x + 6}{(x + 1)\left(x^2 + 4\right)}\,\mathrm{d}x = \ln\left(a\sqrt{2}\right) + b\pi\]where \(a\) and \(b\) are constants to be determined. (4)

Mark scheme (a)

Scheme	Marks	AO
\(\dfrac{2x^2 + 3x + 6}{(x + 1)\left(x^2 + 4\right)} = \dfrac{A}{x + 1} + \dfrac{Bx + C}{x^2 + 4} \Rightarrow 2x^2 + 3x + 6\) \(= A\left(x^2 + 4\right) + (Bx + C)(x + 1)\)	M1	1.1b
e.g. \(x = -1 \Rightarrow A = \ldots,\ x = 0 \Rightarrow C = \ldots,\ \text{coeff } x^2 \Rightarrow B = \ldots\) or Compares coefficients and solves to find values for \(A\), \(B\) and \(C\) \(2 = A + B,\ 3 = B + C,\ 6 = 4A + C\)	dM1	1.1b
\(A = 1,\quad B = 1,\quad C = 2\)	A1	1.1b
	(3)

Notes

M1: Selects the correct form for partial fractions and multiplies through to form suitable identity or uses a method to find at least one value (e.g. cover up rule).

dM1: Full method for finding values for all three constants. Dependent on first M. Allow slips as long as the intention is clear.

A1: Correct constants or partial fractions.

Mark scheme (b)

Scheme	Marks	AO
\(\displaystyle\int_0^2 \frac{1}{x + 1} + \frac{x + 2}{x^2 + 4}\,\mathrm{d}x = \int_0^2 \frac{1}{x + 1} + \frac{x}{x^2 + 4} + \frac{2}{x^2 + 4}\,\mathrm{d}x\) \(= \left[\alpha\ln(x + 1) + \beta\ln\left(x^2 + 4\right) + \lambda\arctan\left(\dfrac{x}{2}\right)\right]_0^2\)	M1	3.1a
\(= \left[\ln(x + 1) + \dfrac{1}{2}\ln\left(x^2 + 4\right) + \arctan\left(\dfrac{x}{2}\right)\right]_0^2\)	A1	2.1
\(= \left[\ln(3) + \dfrac{1}{2}\ln(8) + \arctan 1\right] - \left[\ln(1) + \dfrac{1}{2}\ln(4) + \arctan(0)\right]\) \(= \left[\ln(3) + \dfrac{1}{2}\ln(8) + \arctan(1)\right] - \left[\dfrac{1}{2}\ln 4\right] = \underline{\ln\left(\dfrac{3\sqrt{8}}{2}\right)} + \dfrac{\pi}{4}\)	dM1	2.1
\(\ln\left(3\sqrt{2}\right) + \dfrac{\pi}{4}\)	A1	2.2a
	(4)
	(7 marks)

Notes

M1: Splits the integral into an integrable form and integrates at least two terms to the correct form. They may use a substitution on the arctan term

A1: Fully correct Integration.

dM1: Uses the limits of 0 and 2 (or appropriate for a substitution), subtracts the correct way round and combines the ln terms from separate integrals to a single term with evidence of correct ln laws at least once.

A1: Correct answer