| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.2 This is a straightforward application of the power rule for integration requiring students to rewrite terms with negative exponents and integrate each term separately. It's routine P1/C1 content with no problem-solving element—purely procedural manipulation of standard forms, making it easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(\int\left(\frac{8x^3}{5}-\frac{2}{3x^4}-1\right)dx = \frac{1}{4}\times\frac{8x^4}{5}-\frac{2}{3}\times\frac{1}{-3}x^{-3}-x\) | M1 A1 | For \(x^3 \to x^4\) or \(x^{-4} \to x^{-3}\) or \(1 \to x\). Also allow e.g. \(x^3 \to x^{3+1}\). Any 2 correct unsimplified or simplified terms; indices must be processed but fractions within fractions acceptable. \(+c\) does not count as a correct term here. |
| \(\frac{2}{5}x^4+\frac{2}{9}x^{-3}-x+c\) | A1 A1 | For any 2 correct simplified terms. \(+c\) does not count here. Accept exact decimals e.g. \(0.\dot{2}\). Condone \(-1x\), \(-1x^1\), \(-\frac{x}{1}\), \(+-1x^1\). Final A1: all correct on same line including \(+c\); \(\frac{2}{9x^3}\) acceptable but not \(\frac{\frac{2}{9}}{x^3}\). Condone \(-1x\) only. If any additional/incorrect notation present and no correct expression seen alone, withhold final mark. |
# Question 1:
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $\int\left(\frac{8x^3}{5}-\frac{2}{3x^4}-1\right)dx = \frac{1}{4}\times\frac{8x^4}{5}-\frac{2}{3}\times\frac{1}{-3}x^{-3}-x$ | M1 A1 | For $x^3 \to x^4$ or $x^{-4} \to x^{-3}$ or $1 \to x$. Also allow e.g. $x^3 \to x^{3+1}$. Any 2 correct unsimplified or simplified terms; indices must be processed but fractions within fractions acceptable. $+c$ does not count as a correct term here. |
| $\frac{2}{5}x^4+\frac{2}{9}x^{-3}-x+c$ | A1 A1 | For any 2 correct simplified terms. $+c$ does not count here. Accept exact decimals e.g. $0.\dot{2}$. Condone $-1x$, $-1x^1$, $-\frac{x}{1}$, $+-1x^1$. Final A1: all correct on same line including $+c$; $\frac{2}{9x^3}$ acceptable but not $\frac{\frac{2}{9}}{x^3}$. Condone $-1x$ only. If any additional/incorrect notation present and no correct expression seen alone, withhold final mark. |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( \frac { 8 x ^ { 3 } } { 5 } - \frac { 2 } { 3 x ^ { 4 } } - 1 \right) d x$$
giving each term in simplest form.\\
\hfill \mbox{\textit{Edexcel P1 2022 Q1 [4]}}