October 2021 Paper 2 Q16
16 In this question you must show detailed reasoning.
Find \(\displaystyle\int \frac{x}{1+\sqrt{x}}\,\mathrm{d}x\). [8]
| Scheme | Marks | AO |
|---|---|---|
| \(u = 1 + \sqrt{x}\) | B1 | 3.1a |
| \(\dfrac{\mathrm{d}u}{\mathrm{d}x} = \dfrac{1}{2}x^{-\frac{1}{2}}\) | M1 | 1.1 |
| \(\sqrt{x} = u - 1\) | M1 | 1.1 |
| \(\displaystyle\int \frac{(u-1)^2 \times 2(u-1)}{u}\,\mathrm{d}u\) | A1 | 3.1a |
| \(\dfrac{2(u^3 - 3u^2 + 3u - 1)}{u}\) soi | M1 | 2.1 |
| \(2\left[\dfrac{u^3}{3} - \dfrac{3u^2}{2} + 3u - \ln u\right]\) | M1 A1 | 3.1a 1.1 |
| \(\dfrac{2(1+\sqrt{x})^3}{3} - 3(1+\sqrt{x})^2 + 6(1+\sqrt{x}) - 2\ln(1+\sqrt{x})\) \(+c\) oe isw | A1 | 3.2a |
| [8] |
Notes
B1: for use in substitution
No marks for attempts based solely on integration by parts
M1: allow M1 for \(x^{-\frac{1}{2}}\)
M1: allow sign error
A1: \(\mathrm{d}u\) may be seen later
M1: allow sign errors and/or omission of 2
M1: divides their cubic through by \(u\) and integrates dependent on award of first two M marks;
allow sign errors and coefficient errors
must have \(\ln u\)
A1: if answer fully correct but either \(+\ c\) or \(\mathrm{d}u\) not seen then withhold final A1
Alternative
| Scheme | Marks |
|---|---|
| \(x = u^2\) | B1 |
| \(\dfrac{\mathrm{d}x}{\mathrm{d}u} = 2u\) | M1 |
| \(\sqrt{x} = u\) | M1 |
| \(\displaystyle\int \frac{u^2 \times 2u}{1+u}\,\mathrm{d}u\) | A1 |
| \((2u^2 - 2u + 2) - \dfrac{2}{1+u}\) from long division oe | M1 |
| \(\dfrac{2u^3}{3} - u^2 + 2u - 2\ln(1+u)\) | M1 A1 |
| \(\dfrac{2x\sqrt{x}}{3} - x + 2\sqrt{x} - 2\ln(1+\sqrt{x}) + c\) | A1 |
B1: or \(u = \sqrt{x}\)
may see \(u = \dfrac{1}{1+\sqrt{x}}\) or \(\mathrm{e}^u = 1 + \sqrt{x}\) (corrected from the printed mark scheme: \(\mathrm{e}^u = (1+)\sqrt{x}\))
M1: allow M1 for \(\dfrac{\mathrm{d}u}{\mathrm{d}x} = x^{-\frac{1}{2}}\)
M1: \(x = u^2\)
M1: allow sign errors and/or omission of 2
M1: integration attempted; allow sign errors and coefficient errors
(corrected from the printed mark scheme: the printed line reads \(\dfrac{2u^3}{3} - u^2 + 2 - 2\ln(1+u)\); the term \(2\) should be \(2u\))
A1: if answer fully correct but either \(+\ c\) or \(\mathrm{d}u\) not seen then withhold final A1