| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 5 |
| 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 the power rule for integration requiring students to rewrite terms in index form and apply standard formulas. It's routine practice with no problem-solving element, making it easier than average, though slightly more involved than the simplest recall questions due to requiring manipulation of three different term types (polynomial, root, and negative power). |
| Spec | 1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int 12x^3 + \frac{1}{6\sqrt{x}} - \frac{3}{2x^4}\,dx = 12 \times \frac{x^4}{4} + \frac{1}{6} \times 2x^{\frac{1}{2}} - \frac{3}{2} \times \frac{x^{-3}}{-3}\) | M1 | Applies \(\int x^n\,dx \to x^{n+1}\) for at least one index. Index must be processed: allow \(x^3 \to x^4\) or \(\frac{1}{\sqrt{x}} \to x^{\frac{1}{2}}\) or \(\frac{1}{x^4} \to x^{-3}\) |
| \(= 3x^4 + \frac{1}{3}x^{\frac{1}{2}} + \frac{1}{2}x^{-3} + c\) | A1 | One correct term simplified or \(+c\) |
| A1 | Two correct terms simplified or one correct simplified with \(+c\) | |
| A1 | Three correct terms simplified or two correct simplified with \(+c\) | |
| A1 | All correct and simplified on one line. Allow \(\frac{1}{3}x^{\frac{1}{2}} \leftrightarrow \frac{1}{3}\sqrt{x}\) and \(\frac{1}{2}x^{-3} \leftrightarrow \frac{1}{2x^3}\). Withhold final mark if additional/incorrect notation present |
## Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int 12x^3 + \frac{1}{6\sqrt{x}} - \frac{3}{2x^4}\,dx = 12 \times \frac{x^4}{4} + \frac{1}{6} \times 2x^{\frac{1}{2}} - \frac{3}{2} \times \frac{x^{-3}}{-3}$ | M1 | Applies $\int x^n\,dx \to x^{n+1}$ for at least one index. Index must be processed: allow $x^3 \to x^4$ or $\frac{1}{\sqrt{x}} \to x^{\frac{1}{2}}$ or $\frac{1}{x^4} \to x^{-3}$ |
| $= 3x^4 + \frac{1}{3}x^{\frac{1}{2}} + \frac{1}{2}x^{-3} + c$ | A1 | One correct term simplified or $+c$ |
| | A1 | Two correct terms simplified or one correct simplified with $+c$ |
| | A1 | Three correct terms simplified or two correct simplified with $+c$ |
| | A1 | All correct and simplified on one line. Allow $\frac{1}{3}x^{\frac{1}{2}} \leftrightarrow \frac{1}{3}\sqrt{x}$ and $\frac{1}{2}x^{-3} \leftrightarrow \frac{1}{2x^3}$. Withhold final mark if additional/incorrect notation present |
---\begin{enumerate}
\item Find
\end{enumerate}
$$\int 12 x ^ { 3 } + \frac { 1 } { 6 \sqrt { x } } - \frac { 3 } { 2 x ^ { 4 } } \mathrm {~d} x$$
giving each term in simplest form.\\
\hfill \mbox{\textit{Edexcel P1 2021 Q1 [5]}}