| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule with stated number of strips |
| Difficulty | Standard +0.3 This is a standard C4 integration question combining trapezium rule approximation with integration by parts. Part (a) requires routine application of the trapezium rule formula with straightforward function evaluations. Part (b) is a textbook integration by parts problem (∫x^(1/2)ln(2x)dx), and part (c) involves evaluating definite integrals and simplifying logarithms. While it requires multiple techniques across three parts, each component is standard bookwork with no novel insight required, making it slightly easier than the average A-level question. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.08i Integration by parts1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Table values: \(x=1: \ln 2 = 0.6931\); \(x=2: \sqrt{2}\ln 4 = 1.9605\); \(x=3: \sqrt{3}\ln 6 = 3.1034\); \(x=4: 2\ln 8 = 4.1589\) | M1 | |
| Area \(= \frac{1}{2} \times 1(\ldots)\) | B1 | Correct trapezium rule structure with \(h=1\) |
| \(\approx \ldots(0.6931 + 2(1.9605 + 3.1034) + 4.1589)\) | M1 | Correct bracket structure |
| \(\approx \frac{1}{2} \times 14.97989\ldots \approx 7.49\) | A1 | 7.49 cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int x^{\frac{1}{2}} \ln 2x \, dx = \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \int \frac{2}{3}x^{\frac{3}{2}} \times \frac{1}{x} \, dx\) | M1 A1 | Integration by parts applied correctly |
| \(= \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \int \frac{2}{3}x^{\frac{1}{2}} \, dx\) | ||
| \(= \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \frac{4}{9}x^{\frac{3}{2}} \quad (+C)\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left[\frac{2}{3}x^{\frac{3}{2}} \ln 2x - \frac{4}{9}x^{\frac{3}{2}}\right]_1^4 = \left(\frac{2}{3}\cdot 4^{\frac{3}{2}}\ln 8 - \frac{4}{9}\cdot 4^{\frac{3}{2}}\right) - \left(\frac{2}{3}\ln 2 - \frac{4}{9}\right)\) | M1 | Substituting limits |
| \(= (16\ln 2 - \ldots) - \ldots\) | M1 | Using or implying \(\ln 2^n = n\ln 2\) |
| \(= \frac{46}{3}\ln 2 - \frac{28}{9}\) | A1 |
## Question 7:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Table values: $x=1: \ln 2 = 0.6931$; $x=2: \sqrt{2}\ln 4 = 1.9605$; $x=3: \sqrt{3}\ln 6 = 3.1034$; $x=4: 2\ln 8 = 4.1589$ | M1 | |
| Area $= \frac{1}{2} \times 1(\ldots)$ | B1 | Correct trapezium rule structure with $h=1$ |
| $\approx \ldots(0.6931 + 2(1.9605 + 3.1034) + 4.1589)$ | M1 | Correct bracket structure |
| $\approx \frac{1}{2} \times 14.97989\ldots \approx 7.49$ | A1 | 7.49 cao |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int x^{\frac{1}{2}} \ln 2x \, dx = \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \int \frac{2}{3}x^{\frac{3}{2}} \times \frac{1}{x} \, dx$ | M1 A1 | Integration by parts applied correctly |
| $= \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \int \frac{2}{3}x^{\frac{1}{2}} \, dx$ | | |
| $= \frac{2}{3}x^{\frac{3}{2}} \ln 2x - \frac{4}{9}x^{\frac{3}{2}} \quad (+C)$ | M1 A1 | |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left[\frac{2}{3}x^{\frac{3}{2}} \ln 2x - \frac{4}{9}x^{\frac{3}{2}}\right]_1^4 = \left(\frac{2}{3}\cdot 4^{\frac{3}{2}}\ln 8 - \frac{4}{9}\cdot 4^{\frac{3}{2}}\right) - \left(\frac{2}{3}\ln 2 - \frac{4}{9}\right)$ | M1 | Substituting limits |
| $= (16\ln 2 - \ldots) - \ldots$ | M1 | Using or implying $\ln 2^n = n\ln 2$ |
| $= \frac{46}{3}\ln 2 - \frac{28}{9}$ | A1 | |
---7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{12fbfe89-60fe-4890-9a22-2b1988d05d33-11_754_1177_217_388}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation $y = x ^ { \frac { 1 } { 2 } } \ln 2 x$.\\
The finite region $R$, shown shaded in Figure 3, is bounded by the curve, the $x$-axis and the lines $x = 1$ and $x = 4$
\begin{enumerate}[label=(\alph*)]
\item Use the trapezium rule, with 3 strips of equal width, to find an estimate for the area of $R$, giving your answer to 2 decimal places.
\item Find $\int x ^ { \frac { 1 } { 2 } } \ln 2 x \mathrm {~d} x$.
\item Hence find the exact area of $R$, giving your answer in the form $a \ln 2 + b$, where $a$ and $b$ are exact constants.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2012 Q7 [11]}}