| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Basic indefinite integration |
| Difficulty | Easy -1.3 This is a straightforward C1 integration question requiring only the power rule applied to three terms. It's routine mechanical application with no problem-solving, conceptual understanding, or multi-step reasoning required—simpler than average A-level questions. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(x^n \rightarrow x^{n+1}\) | M1 | Raises any power by 1. E.g. \(x^5 \rightarrow x^6\) or \(x^{-3} \rightarrow x^{-2}\) or \(k \rightarrow kx\). Allow unsimplified e.g. \(x^5 \rightarrow x^{5+1}\) |
| \(2 \times \frac{x^{5+1}}{6}\) or \(-\frac{1}{4} \times \frac{x^{-3+1}}{-2}\) | A1 | Any one of first two terms correct, simplified or unsimplified |
| Two of: \(\frac{1}{3}x^6\), \(\frac{1}{8}x^{-2}\), \(-5x\) | A1 | Any two correct simplified terms. Accept \(+\frac{1}{8x^2}\) for \(+\frac{1}{8}x^{-2}\) but not \(x^1\) for \(x\). Accept \(0.125\) for \(\frac{1}{8}\) but \(\frac{1}{3}\) would need to be identified as \(0.\dot{3}\) |
| \(\frac{1}{3}x^6 + \frac{1}{8}x^{-2} - 5x + c\) | A1 | All correct, simplified, including \(+c\) all on one line. Accept \(+\frac{1}{8x^2}\) for \(+\frac{1}{8}x^{-2}\) but not \(x^1\) for \(x\) |
## Question 1:
$$\int\left(2x^5 - \frac{1}{4}x^{-3} - 5\right)dx$$
| Working/Answer | Mark | Guidance |
|---|---|---|
| $x^n \rightarrow x^{n+1}$ | M1 | Raises any power by 1. E.g. $x^5 \rightarrow x^6$ or $x^{-3} \rightarrow x^{-2}$ or $k \rightarrow kx$. Allow unsimplified e.g. $x^5 \rightarrow x^{5+1}$ |
| $2 \times \frac{x^{5+1}}{6}$ or $-\frac{1}{4} \times \frac{x^{-3+1}}{-2}$ | A1 | Any one of first two terms correct, simplified or unsimplified |
| Two of: $\frac{1}{3}x^6$, $\frac{1}{8}x^{-2}$, $-5x$ | A1 | Any two correct **simplified** terms. Accept $+\frac{1}{8x^2}$ for $+\frac{1}{8}x^{-2}$ but not $x^1$ for $x$. Accept $0.125$ for $\frac{1}{8}$ but $\frac{1}{3}$ would need to be identified as $0.\dot{3}$ |
| $\frac{1}{3}x^6 + \frac{1}{8}x^{-2} - 5x + c$ | A1 | All correct, simplified, including $+c$ all on one line. Accept $+\frac{1}{8x^2}$ for $+\frac{1}{8}x^{-2}$ but not $x^1$ for $x$ |
**Total: 4 marks**
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( 2 x ^ { 5 } - \frac { 1 } { 4 x ^ { 3 } } - 5 \right) \mathrm { d } x$$
giving each term in its simplest form.
\hfill \mbox{\textit{Edexcel C1 2017 Q1 [4]}}