| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| 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 standard integration rules for powers of x, requiring only recall of the formula ∫x^n dx = x^(n+1)/(n+1) + C and basic algebraic simplification. It's a routine P1/AS-level question with no problem-solving element, making it easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Raises any power by 1, e.g. \(x^3 \to x^4\), \(x^{-3} \to x^{-2}\), \(5 \to 5x\) | M1 | Must show attempt at integration |
| \(\frac{2}{3} \times \frac{x^4}{4} - \frac{1}{2} \times \frac{x^{-2}}{-2} + 5x + c\) (un-simplified, two correct terms) | A1 | Accept \(5x^1\) |
| \(\frac{1}{6}x^4 + \frac{1}{4}x^{-2} + 5x\) (two terms correct in simplest form) | A1 | Accept \(\frac{x^4}{6}\), \(\frac{1}{4x^2}\); CONDONE \(+\frac{0.25}{x^2}\); NOT \(\frac{1/4}{x^2}\), \(\frac{5x}{1}\), \(-\left(-\frac{1}{4}x^{-2}\right)\) |
| \(= \frac{1}{6}x^4 + \frac{1}{4}x^{-2} + 5x + c\) fully correct and simplified | A1 | Must have \(+c\); ISW after correct simplified answer |
## Question 1:
$$\int \frac{2}{3}x^3 - \frac{1}{2x^3} + 5 \, dx$$
| Working/Answer | Marks | Guidance |
|---|---|---|
| Raises any power by 1, e.g. $x^3 \to x^4$, $x^{-3} \to x^{-2}$, $5 \to 5x$ | M1 | Must show attempt at integration |
| $\frac{2}{3} \times \frac{x^4}{4} - \frac{1}{2} \times \frac{x^{-2}}{-2} + 5x + c$ (un-simplified, two correct terms) | A1 | Accept $5x^1$ |
| $\frac{1}{6}x^4 + \frac{1}{4}x^{-2} + 5x$ (two terms correct in simplest form) | A1 | Accept $\frac{x^4}{6}$, $\frac{1}{4x^2}$; CONDONE $+\frac{0.25}{x^2}$; NOT $\frac{1/4}{x^2}$, $\frac{5x}{1}$, $-\left(-\frac{1}{4}x^{-2}\right)$ |
| $= \frac{1}{6}x^4 + \frac{1}{4}x^{-2} + 5x + c$ fully correct and simplified | A1 | Must have $+c$; ISW after correct simplified answer |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - \frac { 1 } { 2 x ^ { 3 } } + 5 \right) d x$$
simplifying your answer.\\
\hfill \mbox{\textit{Edexcel P1 2019 Q1 [4]}}