| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Multi-part questions combining substitution with curve/area analysis |
| Difficulty | Standard +0.3 This is a structured multi-part question requiring substitution, integration by parts, and differentiation of logarithms—all standard C3 techniques. The substitution is given explicitly, and part (iii) clearly directs students to use previous results. While it requires careful algebraic manipulation and multiple steps (18 marks total), it follows predictable patterns without requiring novel insight, making it slightly easier than average. |
| Spec | 1.07q Product and quotient rules: differentiation1.08h Integration by substitution1.08i Integration by parts |
\begin{enumerate}[label=(\roman*)]
\item Use the substitution $u = 1 + x$ to show that
$$\int_0^1 \frac{x^3}{1 + x} dx = \int_a^b \left( u^2 - 3u + 3 - \frac{1}{u} \right) du,$$
where $a$ and $b$ are to be found.
Hence evaluate $\int_0^1 \frac{x^3}{1 + x} dx$, giving your answer in exact form.
[7]
Fig. 8 shows the curve $y = x^2 \ln(1 + x)$.
\includegraphics{figure_8}
\item Find $\frac{dy}{dx}$.
Verify that the origin is a stationary point of the curve.
[5]
\item Using integration by parts, and the result of part (i), find the exact area enclosed by the curve $y = x^2 \ln(1 + x)$, the $x$-axis and the line $x = 1$.
[6]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2011 Q8 [18]}}