## Question 16:

| Answer | Mark | Guidance |
|--------|------|----------|
| $u = 1+\sqrt{x}$ | B1 | 3.1a — for use in substitution. No marks for attempts based solely on integration by parts |
| $\frac{\mathrm{d}u}{\mathrm{d}x} = \frac{1}{2}x^{-\frac{1}{2}}$ | M1 | 1.1 — allow M1 for $x^{-\frac{1}{2}}$ |
| $\sqrt{x} = u-1$ | M1 | 1.1 — allow sign error |
| $\int\frac{(u-1)^2 \times 2(u-1)}{u}\ \mathrm{d}u$ | A1 | 3.1a — $\mathrm{d}u$ may be seen later |
| $\frac{2(u^3-3u^2+3u-1)}{u}$ soi | M1 | 2.1 — allow sign errors and/or omission of 2 |
| $2\left[\frac{u^3}{3}-\frac{3u^2}{2}+3u-\ln u\right]$ | M1 | 3.1a — divides their cubic through by $u$ and integrates dependent on award of first two M marks; allow sign errors and coefficient errors; must have $\ln u$ |
| (unsimplified form correct) | A1 | 1.1 |
| $\frac{2(1+\sqrt{x})^3}{3}-3(1+\sqrt{x})^2+6(1+\sqrt{x})-2\ln(1+\sqrt{x})$ $+c$ **oe isw** | A1 | 3.2a — if answer fully correct but either $+c$ or $\mathrm{d}u$ not seen then withhold final A1 |

**Total: [8]**

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## Question 16 (Alternative):

| Answer | Mark | Guidance |
|--------|------|----------|
| $x=u^2$ **or** $u=\sqrt{x}$ | B1 | may also see $u=\frac{1}{1+\sqrt{x}}$ or $e^u=(1+)\sqrt{x}$ |
| $\frac{\mathrm{d}x}{\mathrm{d}u}=2u$ | M1 | allow M1 for $\frac{\mathrm{d}u}{\mathrm{d}x}=x^{-\frac{1}{2}}$ |
| $\sqrt{x}=u$ | M1 | $x=u^2$ |
| $\int\frac{u^2\times 2u}{1+u}\ \mathrm{d}u$ | A1 | |
| $(2u^2-2u+2)-\frac{2}{1+u}$ from long division **oe** | M1 | allow sign errors and/or omission of 2 |
| $\frac{2u^3}{3}-u^2+2-2\ln(1+u)$ | M1 | integration attempted; allow sign errors and coefficient errors |
| (correct) | A1 | |
| $\frac{2x\sqrt{x}}{3}-x+2\sqrt{x}-2\ln(1+\sqrt{x})+c$ | A1 | if answer fully correct but either $+c$ or $\mathrm{d}u$ not seen then withhold final A1 |

**Total: [8]**