## Question 1:

| Working/Answer | Mark | Guidance |
|---|---|---|
| $\frac{2\sqrt{x}}{x^2} \to \ldots x^{-\frac{3}{2}}$ or $\frac{-3}{x^2} \to \ldots x^{-2}$ | M1 | For separating the fraction into two separate terms. Award for one correct index (which does not need to be processed, e.g. $\ldots x^{\frac{1}{2}-2}$). Note $-\frac{3}{x^2}$ is insufficient; must write as $\ldots x^{-2}$ or implied by further work. Beware of candidates who integrate numerator and denominator — incorrect method scores M0A0dM0A0 |
| $\int \frac{2\sqrt{x}-3}{x^2}\,dx = \int 2x^{-\frac{3}{2}} - 3x^{-2}\,dx$ | A1 | $2x^{-\frac{3}{2}} - 3x^{-2}$ o.e. where indices have been processed (may be implied by further work) |
| $\ldots x^{-\frac{3}{2}} \to \ldots x^{-\frac{1}{2}}$ or $\ldots x^{-2} \to \ldots x^{-1}$ | dM1 | For raising the power by one on at least one term with a correct index. Dependent on previous M mark. Index does not need to be processed, e.g. $\ldots x^{-\frac{3}{2}} \to \ldots x^{-\frac{3}{2}+1}$. Not for $+c$ |
| $\int 2x^{-\frac{3}{2}} - 3x^{-2}\,dx = -4x^{-\frac{1}{2}} + 3x^{-1} + c$ | A1 | All correct, simplified, on one line including $+c$. Allow equivalents e.g. $-\frac{4}{\sqrt{x}}$ for $-4x^{-\frac{1}{2}}$ or $\frac{3}{x}$ for $3x^{-1}$. Do not allow e.g. $+-4x^{-\frac{1}{2}}$ or $-\frac{4}{1}x^{-\frac{1}{2}}$. Award once correct expression seen (isw), but withhold final mark if additional/incorrect notation and no correct expression seen on its own. Ignore $y=\ldots$ |
| **Total: 4 marks** | | |

**Alternative method (integration by parts):**

$\int \frac{2\sqrt{x}-3}{x^2}\,dx = \int(2\sqrt{x}-3)x^{-2}\,dx = -(2x^{\frac{1}{2}}-3)x^{-1} + \int x^{-\frac{3}{2}}\,dx$

$-(2x^{\frac{1}{2}}-3)x^{-1} + \int x^{-\frac{3}{2}}\,dx = -2x^{-\frac{1}{2}} + 3x^{-1} - 2x^{-\frac{1}{2}} + c = -4x^{-\frac{1}{2}} + 3x^{-1} + c$