| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Basic indefinite integration |
| Difficulty | Easy -1.2 This is a straightforward C1 integration question requiring only the power rule applied to three terms. Students must rewrite 1/√x as x^(-1/2), integrate each term separately using the standard formula, and simplify. No problem-solving or conceptual insight needed—pure routine application of a basic technique. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x^n \to x^{n+1}\), one power increased by 1 (not just \(+c\)) | M1 | Could be for \(3 \to 3x\) or for \(x^n \to x^{n+1}\) on what they think \(\frac{1}{\sqrt{x}}\) is as a power of \(x\) |
| \(\frac{2}{5}x^5 - \frac{4}{\frac{1}{2}}x^{\frac{1}{2}} + 3x\); one of these 3 terms correct | A1 | Allow un-simplified e.g. \(\frac{2x^{4+1}}{4+1}\), \(-\frac{4x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\), \(3x^1\) |
| Two of these 3 terms correct | A1 | Allow un-simplified as above |
| \(= \frac{2}{5}x^5 - 8x^{\frac{1}{2}} + 3x + c\) | A1 | Complete fully correct simplified expression on one line with constant. Allow \(0.4\) for \(\frac{2}{5}\). Do not allow \(3x^1\) for \(3x\). Allow \(\sqrt{x}\) or \(x^{0.5}\) for \(x^{\frac{1}{2}}\) |
# Question 1:
$$\int \left(2x^4 - \frac{4}{\sqrt{x}} + 3\right)dx$$
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^n \to x^{n+1}$, one power increased by 1 (not just $+c$) | M1 | Could be for $3 \to 3x$ or for $x^n \to x^{n+1}$ on what they think $\frac{1}{\sqrt{x}}$ is as a power of $x$ |
| $\frac{2}{5}x^5 - \frac{4}{\frac{1}{2}}x^{\frac{1}{2}} + 3x$; one of these 3 terms correct | A1 | Allow un-simplified e.g. $\frac{2x^{4+1}}{4+1}$, $-\frac{4x^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}$, $3x^1$ |
| Two of these 3 terms correct | A1 | Allow un-simplified as above |
| $= \frac{2}{5}x^5 - 8x^{\frac{1}{2}} + 3x + c$ | A1 | Complete fully correct simplified expression on one line with constant. Allow $0.4$ for $\frac{2}{5}$. Do not allow $3x^1$ for $3x$. Allow $\sqrt{x}$ or $x^{0.5}$ for $x^{\frac{1}{2}}$ |
**Total: 4 marks**
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int \left( 2 x ^ { 4 } - \frac { 4 } { \sqrt { } x } + 3 \right) d x$$
giving each term in its simplest form.
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\hfill \mbox{\textit{Edexcel C1 2016 Q1 [4]}}