| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2014 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Find indefinite integral of polynomial/power |
| Difficulty | Easy -1.8 This is a straightforward C1 integration question requiring only the basic power rule for polynomials. Students simply add 1 to each power and divide by the new power—pure mechanical application with no problem-solving, making it significantly easier than average A-level questions. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int(8x^3 + 4)dx = \frac{8x^4}{4} + 4x = 2x^4 + 4x + c\) | M1, A1, A1 | Notes: M1 for \(x^n \to x^{n+1}\) so \(x^3 \to x^4\) or \(4 \to 4x\) or \(x^1\). A1 for either term with coefficient unsimplified (power must be simplified)– so \(\frac{8}{4}x^4\) or \(4x\) (accept \(4x^1\)). A1 for fully correct simplified solution with \(c\), i.e. \(2x^4 + 4x + c\) [allow \(2x^4 + 4x + cx^0\)]. If answer given as \(\int 2x^4 + 4x + c\) with integral sign (having never been seen as fully correct simplified answer without integral sign), then give M1A1A0 but allow anything before the \(=\) sign e.g. \(y = 2x^4 + 4x + c\), \(f(x) = 2x^4 + 4x + c\), \(\int = 2x^4 + 4x + c\), etc. If answer followed by (for example) \(x^4 + 2x + k\) then treat as isw (ignore subsequent work). If they follow it by finding a value for \(c\), also isw, provided correct answer with \(c\) has been seen and credited. |
| $\int(8x^3 + 4)dx = \frac{8x^4}{4} + 4x = 2x^4 + 4x + c$ | M1, A1, A1 | **Notes:** M1 for $x^n \to x^{n+1}$ so $x^3 \to x^4$ or $4 \to 4x$ or $x^1$. A1 for either term with coefficient unsimplified (power must be simplified)– so $\frac{8}{4}x^4$ or $4x$ (accept $4x^1$). A1 for fully correct simplified solution with $c$, i.e. $2x^4 + 4x + c$ [allow $2x^4 + 4x + cx^0$]. If answer given as $\int 2x^4 + 4x + c$ with integral sign (having never been seen as fully correct simplified answer without integral sign), then give M1A1A0 but allow anything before the $=$ sign e.g. $y = 2x^4 + 4x + c$, $f(x) = 2x^4 + 4x + c$, $\int = 2x^4 + 4x + c$, etc. If answer followed by (for example) $x^4 + 2x + k$ then treat as isw (ignore subsequent work). If they follow it by finding a value for $c$, also isw, provided correct answer with $c$ has been seen and credited. |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( 8 x ^ { 3 } + 4 \right) d x$$
giving each term in its simplest form.\\
\hfill \mbox{\textit{Edexcel C1 2014 Q1 [3]}}