| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Moderate -0.3 Part (a) is a standard integration by parts application with a simple trigonometric function. Part (b) is a routine substitution that becomes a straightforward logarithm evaluation. Both are textbook exercises requiring direct application of techniques with no problem-solving insight needed, making this slightly easier than average. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\int x\cos 2x\, dx\) | M1 | |
| \(= \frac{x}{2}\sin 2x - \int \frac{1}{2}\sin 2x\, dx\) | A1 | |
| \(= \frac{x}{2}\sin 2x + \frac{1}{4}\cos 2x + c\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(u = x^2 + 1 \Rightarrow \frac{du}{dx} = 2x\) | M1 | Substitution |
| \(x=2 \Rightarrow u=5,\ x=3 \Rightarrow u=10\) | A1 | Limits |
| Integral in \(u\) correct | A1 | Integral in \(u\) |
| \(= \frac{1}{2}\int_5^{10}\frac{1}{u}\,du = \frac{1}{2}[\ln u]_5^{10} = \frac{1}{2}\ln 2\) | A1 | Answer |
| Total: 5 |
## Question 6:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\int x\cos 2x\, dx$ | M1 | |
| $= \frac{x}{2}\sin 2x - \int \frac{1}{2}\sin 2x\, dx$ | A1 | |
| $= \frac{x}{2}\sin 2x + \frac{1}{4}\cos 2x + c$ | A1 | |
| | **Total: 3** | |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $u = x^2 + 1 \Rightarrow \frac{du}{dx} = 2x$ | M1 | Substitution |
| $x=2 \Rightarrow u=5,\ x=3 \Rightarrow u=10$ | A1 | Limits |
| Integral in $u$ correct | A1 | Integral in $u$ |
| $= \frac{1}{2}\int_5^{10}\frac{1}{u}\,du = \frac{1}{2}[\ln u]_5^{10} = \frac{1}{2}\ln 2$ | A1 | Answer |
| | **Total: 5** | |
---6
\begin{enumerate}[label=(\alph*)]
\item Find $\int x \cos 2 x d x$.
\item Using the substitution $u = x ^ { 2 } + 1$, or otherwise, find the exact value of $\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q6 [8]}}