| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 3 |
| 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 integration question requiring only algebraic manipulation (splitting the fraction into two terms) followed by standard power rule integration. The steps are routine: rewrite as (1/2)x^(-1) - (5/2)x^(-3), then integrate to get (1/2)ln|x| + (5/4)x^(-2) + C. No problem-solving insight or reverse chain rule application is actually needed despite the topic label. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(\int \frac{x^2-5}{2x^3}dx = \int Ax^{-1} - Bx^{-3}\,dx = C\ln x + Dx^{-2}(+c)\) | M1 dM1 | M1: Correct attempt to integrate by dividing by \(x^3\) term, forming sum of two terms. Award for one term in correct form: either \(C\ln x+\ldots\) or \(\ldots+Dx^{-2}\). dM1: Achieves both terms in correct form, \(\pm C\ln x \pm Dx^{-2}\) or equivalent |
| \(= \frac{1}{2}\ln x + \frac{5}{4}x^{-2} + c\) | A1 | Must be in simplest form with \(+c\). e.g. \(\ln\sqrt{x}+\frac{5}{4x^2}+c\) also acceptable. Note \(\frac{1}{2}\ln 2x + \frac{5}{4}x^{-2}+c\) is A0 |
# Question 1:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $\int \frac{x^2-5}{2x^3}dx = \int Ax^{-1} - Bx^{-3}\,dx = C\ln x + Dx^{-2}(+c)$ | M1 dM1 | M1: Correct attempt to integrate by dividing by $x^3$ term, forming sum of two terms. Award for one term in correct form: either $C\ln x+\ldots$ **or** $\ldots+Dx^{-2}$. dM1: Achieves both terms in correct form, $\pm C\ln x \pm Dx^{-2}$ or equivalent |
| $= \frac{1}{2}\ln x + \frac{5}{4}x^{-2} + c$ | A1 | Must be in simplest form with $+c$. e.g. $\ln\sqrt{x}+\frac{5}{4x^2}+c$ also acceptable. Note $\frac{1}{2}\ln 2x + \frac{5}{4}x^{-2}+c$ is A0 |
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\item Find
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$$\int \frac { x ^ { 2 } - 5 } { 2 x ^ { 3 } } \mathrm {~d} x \quad x > 0$$
giving your answer in simplest form.\\
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\hfill \mbox{\textit{Edexcel P3 2021 Q1 [3]}}