# Question 6:

## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int \frac{x^3 + 3x^{\frac{1}{2}}}{x}dx = \int\left(x^2 + 3x^{-\frac{1}{2}}\right)dx$ | M1 | Simplify and attempt integration. Need to divide both terms by $x$. Need to simplify each term as far as $x^n$ before integrating. Need to increase power by 1 for at least one term. |
| $= \frac{1}{3}x^3 + 6x^{\frac{1}{2}} + c$ | A1 | Obtain at least one correct term. Allow unsimplified terms. |
| $\frac{1}{3}x^3 + 6x^{\frac{1}{2}}$ | A1 | Obtain $\frac{1}{3}x^3 + 6x^{\frac{1}{2}}$. Coefficients must be simplified. Could be $6\sqrt{x}$ for second term. |
| $+ c$ | B1 | **4 marks** Obtain $+c$. Not dependent on previous marks as long as no longer original function. B0 if integral sign or $dx$ still present. |

## Part (b)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_2^a 6x^{-4}dx = \left[-2x^{-3}\right]_2^a$ | M1 | Obtain integral of the form $kx^{-3}$. Any $k$ numerical. Allow $+c$. Condone integral sign or $dx$ still present. |
| $= \frac{1}{4} - 2a^{-3}$ | M1 | Attempt $F(a) - F(2)$. Must be correct order and subtraction. $-2a^{-3} - \frac{2}{8}$ is M0 unless clear evidence of sign error intention. |
| $\frac{1}{4} - 2a^{-3}$ | A1 | **3 marks** Obtain $\frac{1}{4} - 2a^{-3}$. Allow $\frac{2}{8}$ for $\frac{1}{4}$, but not $-\left(-\frac{2}{8}\right)$. A0 if $+c$, integral sign or $dx$ still present. |

## Part (b)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{4}$ | B1ft | **1 mark** State $\frac{1}{4}$, following their (i). Allow $\frac{2}{8}$. Do not allow $0 + \frac{1}{4}$. Must appreciate that limit is required not inequality. Picking large $a$ and concluding correctly is B1. For ft mark their (i) must be of form $a \pm bx^{-n}$, $n \neq 4$. |

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