FP3 June 2018 Q5
5. Given that \[I_n = \int x^n\sqrt{(x + 8)}\,\mathrm{d}x, \qquad n \geqslant 0,\ x \geqslant 0\]
(a) show that, for \(n \geqslant 1\) \[I_n = \frac{px^n(x + 8)^{\frac{3}{2}}}{2n + 3} - \frac{qn}{2n + 3}I_{n-1}\] where \(p\) and \(q\) are constants to be found. (6)
(b) Use part (a) to find the exact value of \[\int_0^{10}x^2\sqrt{(x + 8)}\,\mathrm{d}x\] giving your answer in the form \(k\sqrt{2}\), where \(k\) is rational. (5)
| Scheme | Marks |
|---|---|
| \(\displaystyle I_n = \int x^n\sqrt{(x + 8)}\,\mathrm{d}x\) | |
| \(\displaystyle I_n = \frac{2}{3}x^n(x + 8)^{\frac{3}{2}} - \int\frac{2}{3}nx^{n-1}(x + 8)^{\frac{3}{2}}(\mathrm{d}x)\) | M1 A1 |
| \(\displaystyle I_n = \ldots - \frac{2}{3}n\int x^{n-1}(x + 8)(x + 8)^{\frac{1}{2}}(\mathrm{d}x)\) | M1 |
| \(I_n = \dfrac{2}{3}x^n(x + 8)^{\frac{3}{2}} - \dfrac{2}{3}nI_n - \dfrac{16}{3}nI_{n-1}\) | dM1 |
| \(I_n + \dfrac{2}{3}nI_n = \dfrac{2}{3}x^n(x + 8)^{\frac{3}{2}} - \dfrac{16}{3}nI_{n-1}\) | ddM1 |
| \(I_n = \dfrac{2x^n(x + 8)^{\frac{3}{2}}}{2n + 3} - \dfrac{16n}{2n + 3}I_{n-1}\) | A1 |
| (6) |
Notes
M1: Parts in the correct direction
A1: Correct expression
M1: Writes \((x + 8)^{\frac{3}{2}}\) as \((x + 8)(x + 8)^{\frac{1}{2}}\)
dM1: Substitutes \(I_n\) and \(I_{n-1}\) correctly. Dependent on the previous M mark
ddM1: Collects \(I_n\) terms to lhs. Dependent on both previous M marks
A1: All correct
| Scheme | Marks |
|---|---|
| \(\displaystyle I_0 = \int\sqrt{(x + 8)}\,\mathrm{d}x = \frac{2}{3}(x + 8)^{\frac{3}{2}}(+c)\) | M1 A1 |
| The first 2 marks may be implied by \(\dfrac{76\sqrt{2}}{3}\) | |
| \(I_2 = \dfrac{2x^2(x + 8)^{\frac{3}{2}}}{2(2) + 3} - \dfrac{16(2)}{2(2) + 3}I_1\) or \(I_1 = \dfrac{2x(x + 8)^{\frac{3}{2}}}{2(1) + 3} - \dfrac{16(1)}{2(1) + 3}I_0\) | M1 |
| \(I_2 = \dfrac{2x^2(x + 8)^{\frac{3}{2}}}{2(2) + 3} - \dfrac{16(2)}{2(2) + 3}I_1\) and \(I_1 = \dfrac{2x(x + 8)^{\frac{3}{2}}}{2(1) + 3} - \dfrac{16(1)}{2(1) + 3}I_0\) \(I_2 = \dfrac{2x^2(x + 8)^{\frac{3}{2}}}{2(2) + 3} - \dfrac{16(2)}{2(2) + 3}\left(\dfrac{2x(x + 8)^{\frac{3}{2}}}{2(1) + 3} - \dfrac{16(1)}{2(1) + 3}I_0\right) = \ldots\) A full complete and correct method with limits applied to obtain a numerical value for \(I_2\) (i.e. there should be no \(x\)’s) Dependent on both previous M marks | ddM1 |
| \(\displaystyle\int_0^{10}x^2\sqrt{(x + 8)}\,\mathrm{d}x = \frac{97232}{105}\sqrt{2}\) | A1 |
| (5) | |
| (11 marks) |
Notes
M1: Attempts \(I_0\) (must be of the form \(k(x + 8)^{\frac{3}{2}}\))
A1: Correct expression
M1: Reduction formula applied at least once
A1: Cao
Useful information
Expression without limits applied: \(I_2 = \dfrac{2x^2(x + 8)^{\frac{3}{2}}}{7} - \dfrac{64x(x + 8)^{\frac{3}{2}}}{35} + \dfrac{1024(x + 8)^{\frac{3}{2}}}{105}\)
This would imply the first 3 marks
Expression with limits applied: \(I_2 = \dfrac{37872}{35}\sqrt{2} - \dfrac{16384}{105}\sqrt{2}\)
Value of \(I_1\): \(I_1 = \dfrac{2024}{15}\sqrt{2}\)
(b) Alternative by parts from scratch
| Scheme | Marks |
|---|---|
| \(\displaystyle I_2 = \int x^2\sqrt{(x + 8)}\,\mathrm{d}x = \frac{2}{3}x^2(8 + x)^{\frac{3}{2}} - \frac{4}{3}\int x(8 + x)^{\frac{3}{2}}\,\mathrm{d}x\) M1: Correct first application of parts on \(I_2\) A1: Correct expression | M1A1 |
| \(\displaystyle = \frac{2}{3}x^2(8 + x)^{\frac{3}{2}} - \frac{4}{3}\left(\frac{2}{5}x(8 + x)^{\frac{5}{2}} - \int\frac{2}{5}(8 + x)^{\frac{5}{2}}\,\mathrm{d}x\right)\) M1: Applies parts again | M1 |
| \(\displaystyle = \frac{2}{3}x^2(8 + x)^{\frac{3}{2}} - \frac{8}{15}x(8 + x)^{\frac{5}{2}} + \frac{8}{15}\int(8 + x)^{\frac{5}{2}}\,\mathrm{d}x\) | |
| \(= \dfrac{2}{3}x^2(8 + x)^{\frac{3}{2}} - \dfrac{8}{15}x(8 + x)^{\frac{5}{2}} + \dfrac{16}{105}(8 + x)^{\frac{7}{2}}\) | |
| \(\left[\dfrac{2}{3}x^2(8 + x)^{\frac{3}{2}} - \dfrac{8}{15}x(8 + x)^{\frac{5}{2}} + \dfrac{16}{105}(8 + x)^{\frac{7}{2}}\right]_0^{10} = \dfrac{200}{3}18^{\frac{3}{2}} - \dfrac{80}{15}18^{\frac{5}{2}} + \dfrac{16}{105}18^{\frac{7}{2}} - \dfrac{16}{105}8^{\frac{7}{2}}\) A fully complete and correct method including correct use of limits to obtain a numerical value for \(I_2\) Dependent on both previous M marks | ddM1 |
| \(= \dfrac{97232}{105}\sqrt{2}\) | A1 |
| (5) |
A1: Cao
Hybrid
| Scheme | Marks |
|---|---|
| \(\displaystyle I_1 = \int x\sqrt{(x + 8)}\,\mathrm{d}x = \frac{2}{3}x(8 + x)^{\frac{3}{2}} - \frac{2}{3}\int(8 + x)^{\frac{3}{2}}\,\mathrm{d}x\) M1: Correct application of parts on \(I_1\) A1: Correct expression | M1A1 |
| \(I_2 = \dfrac{2x^2(x + 8)^{\frac{3}{2}}}{2(2) + 3} - \dfrac{16(2)}{2(2) + 3}I_1\) | M1 |
| \(\displaystyle I_1 = \int x\sqrt{(x + 8)}\,\mathrm{d}x = \frac{2}{3}x(8 + x)^{\frac{3}{2}} - \frac{4}{15}(8 + x)^{\frac{5}{2}}\) | |
| \(I_1 = \left[\dfrac{2}{3}x(8 + x)^{\frac{3}{2}} - \dfrac{4}{15}(8 + x)^{\frac{5}{2}}\right]_0^{10} = \dfrac{2024}{15}\sqrt{2}\) | |
| \(\displaystyle\int_0^{10}x^2\sqrt{(x + 8)}\,\mathrm{d}x = \left[\frac{2x^2(x + 8)^{\frac{3}{2}}}{2(2) + 3}\right]_0^{10} - \frac{32}{7} \times \frac{2024}{15}\sqrt{2} = \ldots\) M1: A complete method including correct use of limits Dependent on both previous M marks | ddM1 |
| \(= \dfrac{97232}{105}\sqrt{2}\) | A1 |
| (5) |
M1: Uses the given reduction formula on \(I_2\)
A1: Cao