A2 October 2020 Q7
7.
\[I_n = \int (4 - x^2)^{-n}\,\mathrm{d}x \qquad n \gt 0\]| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_n = \int 1 \times (4 - x^2)^{-n}\,\mathrm{d}x = x(4 - x^2)^{-n} \pm \int x.n(4 - x^2)^{-n-1}.2x\,\mathrm{d}x\) | M1 | 3.1a |
| \(\displaystyle I_n = x(4 - x^2)^{-n} - \int x.{-n}(4 - x^2)^{-n-1}.{-2x}\,\mathrm{d}x\) | A1 | 1.1b |
| \(\displaystyle = x(4 - x^2)^{-n} + 2n\int (4 - x^2 - 4)(4 - x^2)^{-n-1}\,\mathrm{d}x\) \(\displaystyle = x(4 - x^2)^{-n} + 2n\int (4 - x^2)(4 - x^2)^{-n-1} - 4(4 - x^2)^{-n-1}\,\mathrm{d}x\) | M1 | 2.1 |
| \(\displaystyle I_n = x(4 - x^2)^{-n} + 2n\int (4 - x^2)^{-n}\,\mathrm{d}x - 8n\int (4 - x^2)^{-(n+1)}\,\mathrm{d}x\) \(= x(4 - x^2)^{-n} + 2nI_n - 8nI_{n+1}\) | M1 | 1.1b |
| \(\Rightarrow 8nI_{n+1} = x(4 - x^2)^{-n} + (2n - 1)I_n \Rightarrow I_{n+1} = \dfrac{x}{8n(4 - x^2)^n} + \dfrac{2n - 1}{8n}I_n\) * | A1* | 2.1 |
| (5) |
Notes
M1: Splits integrand as \(1 \times (4 - x^2)^{-n}\) and attempts parts (the right way round). There are lots of minus signs around, so accept with \(\pm\) between as it is hard to tell if the formula is incorrect or it is sign error.
A1: Correct result obtained, need not be simplified.
M1: Gathers the terms and writes the \(x^2\) as \(-(4 - x^2 - 4)\) (oe). Accept either line shown in the scheme (with their coefficients from the parts expansion).
M1: Sorts out the indices and replaces the appropriate integrals by \(I_n\) and \(I_{n+1}\) respectively.
A1*: Completes to the correct printed result, with no errors or ambiguities.
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_1 = \int \frac{1}{4 - x^2}\,\mathrm{d}x = \frac{1}{2}\operatorname{artanh}\left(\frac{x}{2}\right)\) or \(\dfrac{1}{4}\ln\left|\dfrac{2 + x}{2 - x}\right|\) oe | B1 | 2.2a |
| \(I_2 = \dfrac{x}{8(4 - x^2)} + \dfrac{1}{8}I_1\) | M1 | 1.1b |
| \(= \dfrac{x}{8(4 - x^2)} + \dfrac{1}{16}\operatorname{artanh}\left(\dfrac{x}{2}\right)(+c)\) oe e.g. \(\dfrac{x}{8(4 - x^2)} + \dfrac{1}{32}\ln\left|\dfrac{2 + x}{2 - x}\right|(+c)\) | A1 | 1.1b |
| (3) | ||
| (8 marks) |
Notes
B1: Deduces the correct result for \(I_1\)
M1: Applies the reduction formula correct with \(n = 1\) only.
A1: Correct answer, accept any equivalent form as long as fractions are simplified. Constant of integration is not needed.