# Question 4:

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\frac{dx}{du}=2u$ or $\frac{du}{dx}=\frac{1}{2}(x+2)^{-\frac{1}{2}}$ | B1 | Or equivalent correct derivative in any form. May be implied by substitution |
| $\int\frac{(x+2)^{\frac{1}{2}}}{x+5}\,dx=\int\frac{(u^2)^{\frac{1}{2}}}{u^2-2+5}\cdot 2u\,(du)$ | M1 | Complete substitution including their "$dx$". Allow omission of "$du$" if implied by later work |
| $=2\int\frac{u^2}{u^2+3}\,(du)$ or $\int\frac{2u^2}{u^2+3}\,(du)$ | A1 | Correct integral |
| $(2)\int\frac{u^2}{u^2+3}\,du=(2)\int\left(1-\frac{3}{u^2+3}\right)du$ | M1 | Splits the fraction into $A+\frac{B}{u^2+3}$ |
| $=(2)\left[u-\frac{3}{\sqrt{3}}\arctan\frac{u}{\sqrt{3}}\right]$ | A1 A1 | A1: $u$; A1: $-\frac{3}{\sqrt{3}}\arctan\frac{u}{\sqrt{3}}$ |
| $x=-1\Rightarrow u=1,\quad x=7\Rightarrow u=3$ | B1 | Correct limits |
| $=2\left[\left(3-\frac{3}{\sqrt{3}}\cdot\frac{\pi}{3}\right)-\left(1-\frac{3}{\sqrt{3}}\cdot\frac{\pi}{6}\right)\right]$ | M1 | Substitutes $u$ limits correctly into expression of form $\pm\alpha u\pm\beta\arctan(ku)$, $\alpha,\beta\neq0$ and subtracts correctly |
| $=4-\frac{\sqrt{3}}{3}\pi$ | A1 | cao |

**Alternative (last 6 marks):**

| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $u=\sqrt{3}\tan\theta \Rightarrow (2)\int\frac{u^2}{u^2+3}\,du=(2)\int\frac{3\tan^2\theta}{3\tan^2\theta+3}\cdot\sqrt{3}\sec^2\theta\,d\theta$ | M1 | Use of $u=\sqrt{3}\tan\theta$ and complete substitution |
| $=(2\sqrt{3})\int\tan^2\theta\,d\theta=(2\sqrt{3})\int(\sec^2\theta-1)\,d\theta$ | A1A1 | A1: $\theta$; A1: $\tan\theta$ |
| $=(2\sqrt{3})[\tan\theta-\theta]$ | | |
| $u=1\Rightarrow\theta=\frac{\pi}{6},\quad u=3\Rightarrow\theta=\frac{\pi}{3}$ | B1 | Correct limits |
| $=2\sqrt{3}\left[\left(\sqrt{3}-\frac{\pi}{3}\right)-\left(\frac{1}{\sqrt{3}}-\frac{\pi}{6}\right)\right]$ | M1 | Substitutes $\theta$ limits correctly into expression of form $\pm\alpha\tan\theta\pm\beta\theta$, $\alpha,\beta\neq0$ and subtracts correctly |
| $=4-\frac{\sqrt{3}}{3}\pi$ | A1 | cao |

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