| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Double integration by parts |
| Difficulty | Standard +0.8 This requires applying integration by parts twice in succession, which is more demanding than single-application questions. While the technique is standard C4 content, the double application with careful tracking of signs and terms elevates it above average difficulty, though it remains a recognizable textbook exercise without requiring novel insight. |
| Spec | 1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| \(u = x^2, u' = 2x, v' = \sin x, v = -\cos x\) | M1 | |
| \(I = -x^2 \cos x - \int -2x \cos x \, dx = -x^2 \cos x + \int 2x \cos x \, dx\) | A2 | |
| \(u = 2x, u' = 2, v' = \cos x, v = \sin x\) | M1 | |
| \(I = -x^2 \cos x + 2x \sin x - \int 2 \sin x \, dx\) | A1 | |
| \(I = -x^2 \cos x + 2x \sin x + 2 \cos x + c\) | A1 | (6) |
$u = x^2, u' = 2x, v' = \sin x, v = -\cos x$ | M1 |
$I = -x^2 \cos x - \int -2x \cos x \, dx = -x^2 \cos x + \int 2x \cos x \, dx$ | A2 |
$u = 2x, u' = 2, v' = \cos x, v = \sin x$ | M1 |
$I = -x^2 \cos x + 2x \sin x - \int 2 \sin x \, dx$ | A1 |
$I = -x^2 \cos x + 2x \sin x + 2 \cos x + c$ | A1 | **(6)**\begin{enumerate}
\item Use integration by parts to find
\end{enumerate}
$$\int x ^ { 2 } \sin x d x$$
\hfill \mbox{\textit{Edexcel C4 Q1 [6]}}