| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation question requiring students to split the fraction into two terms (3x/2 and -2/x³), then integrate using standard power rule. It's easier than average as it only tests basic algebraic simplification followed by routine integration with no problem-solving insight needed. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\int \frac{3x^4-4}{2x^3}\,dx = \int \frac{3}{2}x - 2x^{-3}\,dx\) | M1 | Attempts to divide to form a sum of terms; implied by two terms with one correct index |
| \(\int \frac{3}{2}x - 2x^{-3}\,dx\) | A1 | Indices must have been processed on both terms; condone spurious notation or lack of integral sign |
| \(= \frac{3}{2} \times \frac{x^2}{2} - 2 \times \frac{x^{-2}}{-2} \quad (+c)\) | dM1 | Full strategy to integrate; look for \(ax^p + bx^q\) where \(p=2\) or \(q=-2\) |
| \(= \frac{3}{4}x^2 + \frac{1}{x^2} + c\) | A1 | Correct answer o.e. such as \(\frac{3}{4}x^2 + x^{-2} + c\) |
# Question 3:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\int \frac{3x^4-4}{2x^3}\,dx = \int \frac{3}{2}x - 2x^{-3}\,dx$ | M1 | Attempts to divide to form a sum of terms; implied by two terms with one correct index |
| $\int \frac{3}{2}x - 2x^{-3}\,dx$ | A1 | Indices must have been processed on both terms; condone spurious notation or lack of integral sign |
| $= \frac{3}{2} \times \frac{x^2}{2} - 2 \times \frac{x^{-2}}{-2} \quad (+c)$ | dM1 | Full strategy to integrate; look for $ax^p + bx^q$ where $p=2$ or $q=-2$ |
| $= \frac{3}{4}x^2 + \frac{1}{x^2} + c$ | A1 | Correct answer o.e. such as $\frac{3}{4}x^2 + x^{-2} + c$ |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \frac { 3 x ^ { 4 } - 4 } { 2 x ^ { 3 } } d x$$
writing your answer in simplest form.
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q3 [4]}}