A2 June 2019 Q5
5.
\[I_n = \int \operatorname{cosec}^n x\,\mathrm{d}x \qquad n \in \mathbb{Z}\]| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_n = \int \operatorname{cosec}^{n-2}x\,\operatorname{cosec}^2 x\,\mathrm{d}x\) \(\displaystyle u = \operatorname{cosec}^{n-2}x, \frac{\mathrm{d}v}{\mathrm{d}x} = \operatorname{cosec}^2 x, \int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x\) | M1 | 2.1 |
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x - (n - 2)\int \operatorname{cosec}^{n-3}x\left(\mathord{-}\operatorname{cosec}\,x\cot x\right)(\mathord{-}\cot x)\,\mathrm{d}x\) | A1 | 1.1b |
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x - (n - 2)\int \operatorname{cosec}^{n-2}x\cot^2 x\,\mathrm{d}x\) | ||
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x - (n - 2)\int \operatorname{cosec}^{n-2}x\left(\operatorname{cosec}^2 x - 1\right)\,\mathrm{d}x\) \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x - (n - 2)I_n + (n - 2)I_{n-2}\) | dM1 | 1.1b |
| \(\displaystyle (n - 1)I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x + (n - 2)I_{n-2}\) | ||
| \(\displaystyle I_n = \frac{(n - 2)}{n - 1}I_{n-2} - \frac{\operatorname{cosec}^{n-2}x\cot x}{n - 1}\) * | A1* | 2.1 |
| (4) |
Notes
For Alt 1 and any other similar approaches marking follows the same pattern
M1: Splits the integrand into the product as shown and begins the process of integration by parts For Alt 2 this requires applying the expression for \(\cot^2 x\) in terms of \(\operatorname{cosec}^2 x\), splitting the integral and setting up the process for integration by parts on the composite term.
A1: Correct expression (for Alt 2 it is for a correct application of parts on their second term)
dM1: Depends on previous M. Applies \(\cot^2 x = \pm 1 \pm \operatorname{cosec}^2 x\) and introduces \(I_n\) and \(I_{n-2}\) (For Alt 2 this is for complete substitution for \(I_n\) and \(I_{n-2}\))
A1*: Completes the proof by making \(I_n\) the subject with no errors seen (but condone minor notational slips). For Alt 1 a clear statement of replacing \(n\) by \(n - 2\) oe should be made.
Alt 1 (a)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_n = \int \operatorname{cosec}^{n+1}x\sin x\,\mathrm{d}x\) (Allow with \(n \pm 1\) in power for M’s) \(\displaystyle u = \operatorname{cosec}^{n+1}x, \frac{\mathrm{d}v}{\mathrm{d}x} = \sin x, \int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x\) | M1 | 2.1 |
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^{n+1}x\cos x - (n + 1)\int \operatorname{cosec}^n x\left(\mathord{-}\operatorname{cosec}\,x\cot x\right)(\mathord{-}\cos x)\,\mathrm{d}x\) | A1 | 1.1b |
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^n x\cot x - (n + 1)\int \operatorname{cosec}^n x\cot^2 x\,\mathrm{d}x\) | ||
| \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^n x\cot x - (n + 1)\int \operatorname{cosec}^n x\left(\operatorname{cosec}^2 x - 1\right)\,\mathrm{d}x\) \(\displaystyle I_n = \mathord{-}\operatorname{cosec}^n x\cot x - (n + 1)I_{n+2} + (n + 1)I_n\) | dM1 | 1.1b |
| \(\displaystyle (n + 1)I_{n+2} = \mathord{-}\operatorname{cosec}^n x\cot x + nI_n\) | ||
| replacing \(n\) by \(n - 2\) gives \(\displaystyle I_n = \frac{(n - 2)}{n - 1}I_{n-2} - \frac{\operatorname{cosec}^{n-2}x\cot x}{n - 1}\) * | A1* | 2.1 |
| (4) |
Alt 2 (a)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_n = \int \operatorname{cosec}^{n-2}x\,\operatorname{cosec}^2 x\,\mathrm{d}x = \int \operatorname{cosec}^{n-2}x\left(1 + \cot^2 x\right)\,\mathrm{d}x\) \(\displaystyle = \int \operatorname{cosec}^{n-2}x\,\mathrm{d}x + \int\left(\operatorname{cosec}^{n-2}x\cot x\right)(\cot x)\,\mathrm{d}x\) \(\displaystyle u = \cot x, \frac{\mathrm{d}v}{\mathrm{d}x} = \operatorname{cosec}^{n-2}x\cot x, \int u\frac{\mathrm{d}v}{\mathrm{d}x}\,\mathrm{d}x = uv - \int v\frac{\mathrm{d}u}{\mathrm{d}x}\,\mathrm{d}x\) | M1 | 2.1 |
| \(\displaystyle I_n = I_{n-2} + \cot x\left(-\frac{\operatorname{cosec}^{n-2}x}{n - 2}\right) - \int\left(-\frac{\operatorname{cosec}^{n-2}x}{n - 2}\right)\left(\mathord{-}\operatorname{cosec}^2 x\right)\,\mathrm{d}x\) | A1 | 1.1b |
| \(\displaystyle (n - 2)I_n = (n - 2)I_{n-2} - \operatorname{cosec}^{n-2}x\cot x - \int \operatorname{cosec}^n x\,\mathrm{d}x\) | ||
| \(\displaystyle (n - 2)I_n = (n - 2)I_{n-2} - \operatorname{cosec}^{n-2}x\cot x - I_n\) | dM1 | 1.1b |
| \(\displaystyle (n - 1)I_n = \mathord{-}\operatorname{cosec}^{n-2}x\cot x + (n - 2)I_{n-2}\) | ||
| \(\displaystyle I_n = \frac{(n - 2)}{n - 1}I_{n-2} - \frac{\operatorname{cosec}^{n-2}x\cot x}{n - 1}\) * | A1* | 2.1 |
| (4) |
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_6 = \frac{4}{5}I_4 - \frac{\operatorname{cosec}^4 x\cot x}{5}\) or \(\displaystyle \big[I_6\big]_{\frac{\pi}{3}}^{\frac{\pi}{2}} = \frac{4}{5}\big[I_4\big]_{\frac{\pi}{3}}^{\frac{\pi}{2}} - \left[\frac{\operatorname{cosec}^4 x\cot x}{5}\right]_{\frac{\pi}{3}}^{\frac{\pi}{2}}\) | M1 | 1.1b |
| \(\displaystyle = \frac{4}{5}\left(\frac{2}{3}I_2 - \frac{\operatorname{cosec}^2 x\cot x}{3}\right) - \frac{\operatorname{cosec}^4 x\cot x}{5}\) or with limits etc | M1 | 1.1b |
| \(\displaystyle \big[I_6\big]_{\frac{\pi}{3}}^{\frac{\pi}{2}} = \frac{8}{15}\big[\mathord{-}\cot x\big]_{\frac{\pi}{3}}^{\frac{\pi}{2}} - \left[\frac{4\operatorname{cosec}^2 x\cot x}{15}\right]_{\frac{\pi}{3}}^{\frac{\pi}{2}} - \left[\frac{\operatorname{cosec}^4 x\cot x}{5}\right]_{\frac{\pi}{3}}^{\frac{\pi}{2}}\) | M1 | 2.1 |
| \(\displaystyle = \frac{8}{15}\left(\frac{\sqrt{3}}{3}\right) + \frac{16\sqrt{3}}{135} + \frac{16\sqrt{3}}{135} = \frac{56}{135}\sqrt{3}\) * | A1* | 2.2a |
| (4) | ||
| (8 marks) |
Notes
M1: Begins process of application of reduction to find \(I_6\) in terms of \(I_4\) (need not evaluate terms) or deduces the value of \(I_2\) (Alt 1)
M1: Uses the reduction formula correctly to find \(I_4\) in terms of \(I_2\) (need not be evaluated yet).
M1: A fully correct method using the reduction formula correctly to reach a value for \(I_6\). Substitutions must be shown for the non-zero terms but accept decimals/trig functions for the M.
A1*: Reaches the printed answer with no errors, relevant working shown and trig terms evaluated
Alt (b)
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle I_2 = \int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \operatorname{cosec}^2 x\,\mathrm{d}x = \big[\mathord{-}\cot x\big]_{\frac{\pi}{3}}^{\frac{\pi}{2}} = \frac{\sqrt{3}}{3}\) | M1 | 2.2a |
| \(\displaystyle I_4 = \frac{2}{3}I_2 - \left[\frac{\operatorname{cosec}^2 x\cot x}{3}\right]_{\frac{\pi}{3}}^{\frac{\pi}{2}} = \frac{2}{9}\sqrt{3} + \frac{4}{27}\sqrt{3}\) | M1 | 1.1b |
| \(\displaystyle I_6 = \frac{4}{5}I_4 - \left[\frac{\operatorname{cosec}^4 x\cot x}{5}\right]_{\frac{\pi}{3}}^{\frac{\pi}{2}} = \frac{4}{5}\left(\frac{4}{27}\sqrt{3} + \frac{2}{9}\sqrt{3}\right) + \frac{16}{135}\sqrt{3}\) | M1 | 2.1 |
| \(\displaystyle = \frac{56}{135}\sqrt{3}\) * | A1* | 1.1b |
| (4) |