| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after expanding or multiplying out |
| Difficulty | Moderate -0.8 This is a straightforward integration question requiring only algebraic expansion of the numerator followed by term-by-term integration using the power rule. The manipulation is routine (multiply out brackets, divide by 3) and the integration itself involves standard power rule application with fractional indices. No problem-solving insight or reverse chain rule is actually needed despite the topic label. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^{\frac{1}{2}}(2x-5) = ...x^{\frac{3}{2}} + ...x^{\frac{1}{2}}\) or \(\frac{x^{\frac{1}{2}}(2x-5)}{3} = \frac{...x^{\frac{3}{2}}+...x^{\frac{1}{2}}}{3}\) | M1 | Attempts to multiply out brackets, writing expression as sum of terms with indices. Award for either one correct index. |
| \(\frac{2x^{\frac{3}{2}}}{3} - \frac{5x^{\frac{1}{2}}}{3}\) | A1 | Or equivalent e.g. \(\frac{1}{3}(2x^{\frac{3}{2}}-5x^{\frac{1}{2}})\). Coefficients must be exact. |
| \(\int \frac{2x^{\frac{3}{2}}}{3} - \frac{5x^{\frac{1}{2}}}{3}\, dx = ...x^{\frac{5}{2}} \pm ...x^{\frac{3}{2}}\ (+c)\) | dM1 | Increases power by one on \(x^n\) term where \(n\) is a fraction. Dependent on previous M mark. |
| \(\frac{4}{15}x^{\frac{5}{2}} - \frac{10}{9}x^{\frac{3}{2}} + c\) | A1 | Must include \(+c\). Fractions in lowest terms, indices processed. Accept equivalents e.g. \(\frac{4}{15}\sqrt{x^5}-\frac{10}{9}\sqrt{x^3}+c\). Do not accept e.g. \(0.267x^{\frac{5}{2}}-1.11x^{\frac{3}{2}}+c\). |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^{\frac{1}{2}}(2x-5) = ...x^{\frac{3}{2}} + ...x^{\frac{1}{2}}$ or $\frac{x^{\frac{1}{2}}(2x-5)}{3} = \frac{...x^{\frac{3}{2}}+...x^{\frac{1}{2}}}{3}$ | M1 | Attempts to multiply out brackets, writing expression as sum of terms with indices. Award for either one correct index. |
| $\frac{2x^{\frac{3}{2}}}{3} - \frac{5x^{\frac{1}{2}}}{3}$ | A1 | Or equivalent e.g. $\frac{1}{3}(2x^{\frac{3}{2}}-5x^{\frac{1}{2}})$. Coefficients must be exact. |
| $\int \frac{2x^{\frac{3}{2}}}{3} - \frac{5x^{\frac{1}{2}}}{3}\, dx = ...x^{\frac{5}{2}} \pm ...x^{\frac{3}{2}}\ (+c)$ | dM1 | Increases power by one on $x^n$ term where $n$ is a fraction. Dependent on previous M mark. |
| $\frac{4}{15}x^{\frac{5}{2}} - \frac{10}{9}x^{\frac{3}{2}} + c$ | A1 | Must include $+c$. Fractions in lowest terms, indices processed. Accept equivalents e.g. $\frac{4}{15}\sqrt{x^5}-\frac{10}{9}\sqrt{x^3}+c$. Do not accept e.g. $0.267x^{\frac{5}{2}}-1.11x^{\frac{3}{2}}+c$. |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \frac { x ^ { \frac { 1 } { 2 } } ( 2 x - 5 ) } { 3 } \mathrm {~d} x$$
writing each term in simplest form.
\hfill \mbox{\textit{Edexcel Paper 1 2023 Q1 [4]}}