| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| 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 -1.3 This is a straightforward application of the power rule for integration to three simple terms. It requires only direct recall of the formula ∫x^n dx = x^(n+1)/(n+1) + C with no problem-solving, making it easier than average and typical of basic P1/C1 integration exercises. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int 10x^5 + 6x^3 - \frac{3}{x^2}\,dx = 10 \times \frac{x^6}{6} + 6 \times \frac{x^4}{4} - 3 \times \frac{x^{-1}}{-1}(+c)\) | M1 | Raising any power by 1, e.g. \(x^5 \to x^6\), \(x^3 \to x^4\) or \(x^{-2} \to x^{-1}\). Accept unprocessed indices such as \(x^{5+1}\) |
| At least two of three terms correctly integrated (unsimplified acceptable) | A1 | Accept terms like \(10 \times \frac{x^6}{6}\) and \(-3 \times \frac{x^{-1}}{-1}\). Do NOT accept unprocessed terms like \(10 \times \frac{x^{5+1}}{5+1}\) |
| Two correct and simplified terms on one line | A1 | Accept equivalent terms such as \(3x^{-1}\) for \(\frac{3}{x}\) and \(1.5x^4\) for \(\frac{3x^4}{2}\). Do NOT accept \(-3 \times \frac{x^{-1}}{-1}\) or \(1.67x^6\) |
| \(= \frac{5x^6}{3} + \frac{3x^4}{2} + \frac{3}{x} + c\) | A1 | Fully correct and simplified with \(+c\) all on one line. Do NOT accept with spurious symbols like \(\int(\cdots)dx\) or if they multiply to remove fractions |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int 10x^5 + 6x^3 - \frac{3}{x^2}\,dx = 10 \times \frac{x^6}{6} + 6 \times \frac{x^4}{4} - 3 \times \frac{x^{-1}}{-1}(+c)$ | M1 | Raising any power by 1, e.g. $x^5 \to x^6$, $x^3 \to x^4$ or $x^{-2} \to x^{-1}$. Accept unprocessed indices such as $x^{5+1}$ |
| At least two of three terms correctly integrated (unsimplified acceptable) | A1 | Accept terms like $10 \times \frac{x^6}{6}$ and $-3 \times \frac{x^{-1}}{-1}$. Do NOT accept unprocessed terms like $10 \times \frac{x^{5+1}}{5+1}$ |
| Two correct and simplified terms on one line | A1 | Accept equivalent terms such as $3x^{-1}$ for $\frac{3}{x}$ and $1.5x^4$ for $\frac{3x^4}{2}$. Do NOT accept $-3 \times \frac{x^{-1}}{-1}$ or $1.67x^6$ |
| $= \frac{5x^6}{3} + \frac{3x^4}{2} + \frac{3}{x} + c$ | A1 | Fully correct and simplified with $+c$ all on one line. Do NOT accept with spurious symbols like $\int(\cdots)dx$ or if they multiply to remove fractions |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( 10 x ^ { 5 } + 6 x ^ { 3 } - \frac { 3 } { x ^ { 2 } } \right) \mathrm { d } x$$
giving your answer in simplest form.\\
\hfill \mbox{\textit{Edexcel P1 2022 Q1 [4]}}