| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.8 This is a straightforward P1 integration question requiring only algebraic simplification (splitting the fraction into 2x³ + 3/(2x²)) followed by standard power rule integration. The manipulation is routine and the integration is direct application of formulas with no problem-solving insight needed. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int \frac{4x^5+3}{2x^2}\,dx = \int 2x^3 + \frac{3}{2}x^{-2}\,dx\) | M1 A1 | M1: Attempts to write as a sum of two terms with one processed index correct. Award for \(Px^3 + Qx^k\) or \(Px^k + Qx^{-2}\) or \(Px^k + \frac{Q}{x^2}\) where \(k\) could be 0. A1: Correct integrand as sum of two terms e.g. \(2x^3 + \frac{3}{2}x^{-2}\) seen in one expression. |
| \(= \frac{1}{2}x^4 - \frac{3}{2}x^{-1} + c\) | dM1 A1 A1 | dM1: Attempts to integrate expression of form \(Px^m + Qx^n\), \(m=3\) or \(n=-2\), raising one index by one. Depends on first M. A1: Either term correct and simplified from correct work: \(\frac{1}{2}x^4 + Ax^n(+c)\) or \(Bx^m - \frac{3}{2}x^{-1}(+c)\). A1: \(\frac{1}{2}x^4 - \frac{3}{2}x^{-1} + c\); allow simplified equivalents e.g. \(\frac{1}{2}x^4 - \frac{3}{2x} + c\), \(0.5x^4 - 1.5x^{-1}+c\). Not e.g. \(\frac{1}{2}x^4 - \frac{3}{2} \div x + c\) or \(\frac{1}{2}x^4 + {-\frac{3}{2}}x^{-1}+c\). |
# Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int \frac{4x^5+3}{2x^2}\,dx = \int 2x^3 + \frac{3}{2}x^{-2}\,dx$ | M1 A1 | M1: Attempts to write as **a sum of two terms** with one processed index correct. Award for $Px^3 + Qx^k$ or $Px^k + Qx^{-2}$ or $Px^k + \frac{Q}{x^2}$ where $k$ could be 0. A1: Correct integrand as sum of two terms e.g. $2x^3 + \frac{3}{2}x^{-2}$ seen in one expression. |
| $= \frac{1}{2}x^4 - \frac{3}{2}x^{-1} + c$ | dM1 A1 A1 | dM1: Attempts to integrate expression of form $Px^m + Qx^n$, $m=3$ or $n=-2$, raising one index by one. Depends on first M. A1: Either term correct and simplified from correct work: $\frac{1}{2}x^4 + Ax^n(+c)$ or $Bx^m - \frac{3}{2}x^{-1}(+c)$. A1: $\frac{1}{2}x^4 - \frac{3}{2}x^{-1} + c$; allow simplified equivalents e.g. $\frac{1}{2}x^4 - \frac{3}{2x} + c$, $0.5x^4 - 1.5x^{-1}+c$. **Not** e.g. $\frac{1}{2}x^4 - \frac{3}{2} \div x + c$ or $\frac{1}{2}x^4 + {-\frac{3}{2}}x^{-1}+c$. |\begin{enumerate}
\item Find
\end{enumerate}
$$\int \frac { 4 x ^ { 5 } + 3 } { 2 x ^ { 2 } } \mathrm {~d} x$$
giving your answer in simplest form.
\hfill \mbox{\textit{Edexcel P1 2023 Q3 [5]}}