C4 June 2018 Q3

Notes

M1: Writes down a correct identity (although this can be implied) and attempts to find the value of at least one of either \(A\) or \(B\) or \(C\). This can be achieved by either substituting values into their identity or comparing coefficients.

Note: The correct partial fraction from no working scores B1B1M1A1

Notes

M1: At least 2 of either \(\pm\dfrac{P}{(2x + 1)} \to \pm D\ln(2x + 1)\) or \(\pm D\ln(x + \frac{1}{2})\) or \(\pm\dfrac{Q}{(2x + 1)^2} \to \pm E(2x + 1)^{-1}\)
or \(\pm\dfrac{R}{(x + 3)} \to \pm F\ln(x + 3)\) for their constants \(P\), \(Q\), \(R\).

A1ft: At least two terms from any of \(\pm\dfrac{P}{(2x + 1)}\) or \(\pm\dfrac{Q}{(2x + 1)^2}\) or \(\pm\dfrac{R}{(x + 3)}\) correctly integrated.

Note: Can be un-simplified for the A1ft mark.

A1: Correct answer of \(\dfrac{(-2)}{2}\ln(2x + 1) + \dfrac{6(2x + 1)^{-1}}{(-1)(2)} + \ln(x + 3)\ \{+ c\}\) simplified or un-simplified.
with or without ‘\(+ c\)’.

Note: Allow final A1 for equivalent answers, e.g. \(\ln\left(\dfrac{x + 3}{2x + 1}\right) - \dfrac{3}{2x + 1}\ \{+ c\}\) or \(\ln\left(\dfrac{2x + 6}{2x + 1}\right) - \dfrac{3}{2x + 1}\ \{+ c\}\)

Note: Beware that \(\displaystyle\int \frac{-2}{(2x + 1)}\,\mathrm{d}x = \int \frac{-1}{(x + \frac{1}{2})}\,\mathrm{d}x = -\ln(x + \tfrac{1}{2})\ \{+ c\}\) is correct integration

Note: E.g. Allow M1 A1ft A1 for a correct un-simplified \(\ln(x + 3) - \ln(x + \frac{1}{2}) - \frac{3}{2}(x + \frac{1}{2})^{-1}\ \{+ c\}\)

Note: Condone 1^st A1ft for poor bracketing, but do not allow poor bracketing for the final A1
E.g. Give final A0 for \(-\ln 2x + 1 - 3(2x + 1)^{-1} + \ln x + 3\ \{+ c\}\) unless recovered

Notes

Note: Give B1 for an un-simplified \(\mathrm{e}^{3x} + 2\mathrm{e}^{2x} + \mathrm{e}^{2x} + 2\mathrm{e}^x + \mathrm{e}^x + 1\)

M1: At least 3 of either \(\alpha\mathrm{e}^{3x} \to \dfrac{\alpha}{3}\mathrm{e}^{3x}\) or \(\beta\mathrm{e}^{2x} \to \dfrac{\beta}{2}\mathrm{e}^{2x}\) or \(\delta\mathrm{e}^x \to \delta\mathrm{e}^x\) or \(\mu \to \mu x;\ \alpha, \beta, \delta, \mu \neq 0\)

Note: Give A1 for an un-simplified \(\dfrac{1}{3}\mathrm{e}^{3x} + \mathrm{e}^{2x} + \dfrac{1}{2}\mathrm{e}^{2x} + 2\mathrm{e}^x + \mathrm{e}^x + x\), with or without \(+c\)

Alt 1 for part (ii)

Alt 2 for part (ii)

Notes

Note: 1^st M1 can be implied by \(\displaystyle\int \frac{\pm ku}{4u^2 \pm 5}\{\mathrm{d}u\},\ k \neq 0\). Does not have to include integral sign or \(\mathrm{d}u\)

Note: Condone 1^st M1 for expressions of the form \(\displaystyle\int \left(\frac{\pm 1}{4u^3 \pm 5u}.\frac{\pm k}{u^{-2}}\right)\{\mathrm{d}u\},\ k \neq 0\)

Note: Give 2^nd M0 for \(\dfrac{3u}{8u}\ln(4u^2 + 5)\ \{+ c\}\) (\(u\)’s not cancelled) unless recovered in later working

Note: E.g. Give 2^nd M0 for integration leading to \(\dfrac{3}{4}u\ln(4u^2 + 5)\) as this is not in the form \(\pm\lambda\ln(4u^2 + 5)\)

Note: Condone 2^nd M1 for poor bracketing, but do not allow poor bracketing for the final A1
E.g. Give final A0 for \(\dfrac{3}{8}\ln 4x^{\frac{2}{3}} + 5\ \{+c\}\) unless recovered

Alternative method 1 for part (iii)