| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Use trig identity before indefinite integration |
| Difficulty | Moderate -0.8 This question requires expanding the squared bracket to get standard trigonometric terms, then applying the double angle formula for cos²x and integrating standard forms. It's a routine multi-step question testing algebraic manipulation and standard integral knowledge, but more straightforward than average A-level questions since the techniques are well-practiced and the path is clear once expanded. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int(2\cos x - \sin x)^2\,dx = \int\left(4\cos^2 x - 4\sin x\cos x + \sin^2 x\right)dx\) | M1 | Expands to form \(p\cos^2 x + q\sin x\cos x + r\sin^2 x\) |
| \(\int 4\sin x\cos x\,dx = \int 2\sin 2x\,dx = -\cos 2x\) or \(-2\cos^2 x\) or \(2\sin^2 x\) | M1 | Correct strategy for integrating \(q\sin x\cos x\), obtaining \(k\cos 2x\) or \(k\sin^2 x\) or \(k\cos^2 x\) |
| \(\int\left(4\cos^2 x+\sin^2 x\right)dx = \int\left(1+3\cos^2 x\right)dx = \int\left(1+3\left(\frac{\cos 2x+1}{2}\right)\right)dx\) | M1 | Correct strategy for rewriting \(p\cos^2 x + r\sin^2 x\) into a form that can be integrated. Uses \(\sin^2 x = \frac{\pm1\pm\cos 2x}{2}\) or \(\cos^2 x = \frac{\pm1\pm\cos 2x}{2}\) |
| \(\int(2\cos x-\sin x)^2\,dx = \frac{3}{4}\sin 2x + \cos 2x + \frac{5}{2}x\ (+c)\) | A1 | Integrates and achieves 2 correct terms (of 3 required terms). Unsimplified expressions acceptable |
| or \(\frac{3}{4}\sin 2x + 2\cos^2 x + \frac{5}{2}x\ (+c)\) or \(\frac{3}{4}\sin 2x - 2\sin^2 x + \frac{5}{2}x\ (+c)\) | A1 | Correct simplified integration (\(+c\) not required) |
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int(2\cos x - \sin x)^2\,dx = \int\left(4\cos^2 x - 4\sin x\cos x + \sin^2 x\right)dx$ | M1 | Expands to form $p\cos^2 x + q\sin x\cos x + r\sin^2 x$ |
| $\int 4\sin x\cos x\,dx = \int 2\sin 2x\,dx = -\cos 2x$ or $-2\cos^2 x$ or $2\sin^2 x$ | M1 | Correct strategy for integrating $q\sin x\cos x$, obtaining $k\cos 2x$ or $k\sin^2 x$ or $k\cos^2 x$ |
| $\int\left(4\cos^2 x+\sin^2 x\right)dx = \int\left(1+3\cos^2 x\right)dx = \int\left(1+3\left(\frac{\cos 2x+1}{2}\right)\right)dx$ | M1 | Correct strategy for rewriting $p\cos^2 x + r\sin^2 x$ into a form that can be integrated. Uses $\sin^2 x = \frac{\pm1\pm\cos 2x}{2}$ or $\cos^2 x = \frac{\pm1\pm\cos 2x}{2}$ |
| $\int(2\cos x-\sin x)^2\,dx = \frac{3}{4}\sin 2x + \cos 2x + \frac{5}{2}x\ (+c)$ | A1 | Integrates and achieves 2 correct terms (of 3 required terms). Unsimplified expressions acceptable |
| or $\frac{3}{4}\sin 2x + 2\cos^2 x + \frac{5}{2}x\ (+c)$ or $\frac{3}{4}\sin 2x - 2\sin^2 x + \frac{5}{2}x\ (+c)$ | A1 | Correct simplified integration ($+c$ not required) |\begin{enumerate}
\item Find, in simplest form,
\end{enumerate}
$$\int ( 2 \cos x - \sin x ) ^ { 2 } d x$$
\hfill \mbox{\textit{Edexcel P3 2023 Q8 [5]}}