## Question 8:

### Part 8(a):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left\{\frac{dy}{dx} =\right\} \operatorname{arcosh}(5x) + \frac{ax}{\sqrt{bx^2-1}}$ or $\operatorname{arcosh}(5x) + \frac{cx}{\sqrt{x^2-d}}$ $\Rightarrow \operatorname{arcosh}(5x) + \frac{5x}{\sqrt{25x^2-1}}$ | M1 | Differentiates to obtain expression of correct form $a, b, c, d \neq 0$ |
| $\operatorname{arcosh}(5x) + \frac{5x}{\sqrt{25x^2-1}}$ | A1 | Correct differentiation. Any equivalent form |

**Total: 2 marks**

### Part 8(b):

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{d}{dx}(x\operatorname{arcosh}(5x)) = \operatorname{arcosh}(5x) + \frac{5x}{\sqrt{25x^2-1}}$ $\Rightarrow \int \operatorname{arcosh}(5x)\, dx = x\operatorname{arcosh}(5x) - \int \frac{5x}{\sqrt{25x^2-1}}\, dx$ | M1 | Rearranges answer to (a) correctly and integrates, or uses correct formula to apply parts to $1 \times \operatorname{arcosh}(5x)$ |
| $\int \operatorname{arcosh}(5x)\, dx = x\operatorname{arcosh}(5x) - \int \frac{5x}{\sqrt{25x^2-1}}\, dx$ | A1 (limited ft) | Correct expression |
| $= x\operatorname{arcosh}(5x) - \frac{1}{5}(25x^2-1)^{\frac{1}{2}}\ (+c)$ | M1 A1 (limited ft) | M1: $\int \frac{Ax}{\sqrt{Bx^2-1}}\, dx \to C(Bx^2-1)^{\frac{1}{2}}$. A1: Fully correct expression with $x\operatorname{arcosh}(5x)$ |
| $\int_{\frac{1}{4}}^{\frac{3}{5}} \operatorname{arcosh}(5x)\, dx = \frac{3}{5}\operatorname{arcosh}(3) - \frac{1}{5}\sqrt{25\times\frac{9}{25}-1} - \left(\frac{1}{4}\operatorname{arcosh}\left(\frac{5}{4}\right) - \frac{1}{5}\sqrt{25\times\frac{1}{16}-1}\right)$ | M1 | Applies appropriate limits with subtraction the right way round, to obtain expression of form $x\operatorname{arcosh}(5x) \pm f(x)$ where $f(x)$ has come from integration |
| $= \frac{3}{5}\operatorname{arcosh}(3) - \frac{2\sqrt{2}}{5} - \frac{1}{4}\operatorname{arcosh}\left(\frac{5}{4}\right) + \frac{3}{20}$ | A1 | Correct answer seen in any form. Must not follow clearly incorrect work |
| $\operatorname{arcosh}(3) = \ln(3+\sqrt{3^2-1^2})$ or $\operatorname{arcosh}\left(\frac{5}{4}\right) = \ln\left(\frac{5}{4}+\sqrt{\left(\frac{5}{4}\right)^2-1^2}\right)$ $\Rightarrow \frac{3}{5}\ln(3+\sqrt{8}) - \frac{2\sqrt{2}}{5} - \frac{1}{4}\ln 2 + \frac{3}{20}$ | M1 | Converts $\operatorname{arcosh}(3)$ or $\operatorname{arcosh}\left(\frac{5}{4}\right)$ to any correct log form. Independent mark but must have obtained $x\operatorname{arcosh}(5x) \pm f(x)$ where $f(x)$ has come from integration |
| $= \frac{3}{20} - \frac{2\sqrt{2}}{5} + \ln\left(3+2\sqrt{2}\right)^{\frac{3}{5}} - \frac{1}{4}\ln 2$ | A1 | Correct answer. Terms in any order but otherwise written as shown. Allow values for $p, q, r$ & $k$. Must not follow clearly incorrect work |

**Total: 8 marks**

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