June 2023 Paper 1 Q12
12
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{d}u = \mathrm{e}^x\,\mathrm{d}x\) | B1 | 1.1 |
| \(\displaystyle\int \frac{7(u + 2) - 8}{u^2} \cdot \frac{1}{u + 2}\,\mathrm{d}u\) | M1 | 1.1 |
| \(\displaystyle = \int \frac{7u + 14 - 8}{u^2(u + 2)}\,\mathrm{d}u = \int \frac{7u + 6}{u^2(u + 2)}\,\mathrm{d}u\) | A1 | 2.1 |
| [3] |
Notes
B1: Correct statement linking \(\mathrm{d}u\) and \(\mathrm{d}x\)
or \(\mathrm{d}x = \dfrac{1}{u + 2}\,\mathrm{d}u\)
M1: Use \(\mathrm{e}^x = u + 2\) to attempt integrand in terms of \(u\)
Must see clear evidence of substitution, including how \(\mathrm{e}^x\mathrm{d}x\) is dealt with
M0 for going straight from \(7\mathrm{e}^x - 8\) to \(7u + 6\) with no justification
Must include \(\mathrm{d}u\)
A1: Correct integrand
Including both integral sign and \(\mathrm{d}u\) throughout, as AG
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{A}{u} + \dfrac{B}{u^2} + \dfrac{C}{u + 2} = \dfrac{7u + 6}{u^2(u + 2)}\) \(Au(u + 2) + B(u + 2) + Cu^2 = 7u + 6\) | M1 | 3.1a |
| \(\dfrac{2}{u} + \dfrac{3}{u^2} - \dfrac{2}{u + 2}\) | A1 | 2.1 |
| \(2\ln|u| - 2\ln|u + 2| - 3u^{-1}\) | M1 | 1.1 |
| A1FT | 2.1 | |
| \(\left(2\ln 4 - 2\ln 6 - \frac{3}{4}\right) - \left(2\ln 2 - 2\ln 4 - \frac{3}{2}\right)\) | M1 | 1.1a |
| \(\left(\dfrac{3}{2} - \dfrac{3}{4}\right) + \ln\left(\dfrac{4 \times 4}{6 \times 2}\right)^2\) | M1 | 3.1a |
| \(\dfrac{3}{4} + \ln\dfrac{16}{9}\) | A1 | 2.1 |
| [7] |
Notes
M1: Attempt correct partial fractions
May have \(\dfrac{Au + B}{u^2} + \dfrac{C}{u + 2}\) but M0 for just \(\dfrac{B}{u^2}\) with no \(\dfrac{A}{u}\)
Correct method to combine correct fractions, and at least one constant attempted
If considering \(\dfrac{7}{u^2} + \dfrac{-8}{u^2(u + 2)}\) then must use partial fractions on the second term to get credit
A1: Correct partial fractions
May have \(\dfrac{2u + 3}{u^2} - \dfrac{2}{u + 2}\)
Possibly implied by their \(A\), \(B\), and \(C\) values ie \(A = 2\), \(B = 3\), \(C = -2\)
M1: Attempt integration of \(\dfrac{B}{u^2}\) and at least one of \(\dfrac{A}{u}\) or \(\dfrac{C}{u + 2}\), and no others
Allow errors in coefficients only
Allow M1 if only two fractions, as long as of required form
If using \(\dfrac{Au + B}{u^2}\) then it must be a correct integration attempt (ie split into two fractions first)
A1FT: FT on their two or three fractions as long as \(ku^{-2}\) and one or two fractions each with a linear denominator
Condone brackets not modulus
Condone no brackets as long as implied by later working, eg when limits are used
M1: Attempt use of correct limits – correct order and subtraction; \(u\) or \(x\) but commensurate with their integral
Allow substitution into any function that is clearly attempt at integration
M1: Attempt to rearrange correct numerical integral to required form
Must be correct numerical expression from correct working
Terms may have been combined before use of limits, but must still be correct expression to gain M1
Correct attempt to combine ln terms ie deal with coefficients and correct product / quotient for the sum / differences
Allow one slip
A1: Obtain \(\dfrac{3}{4} + \ln\dfrac{16}{9}\)
Condone \(\dfrac{3}{4} + 2\ln\dfrac{4}{3}\)
Fractions must be simplified
ISW an incorrect attempt to write this answer in a different form, but A0 if further work done eg multiplying by a constant to clear the fractions