| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Show definite integral equals value |
| Difficulty | Moderate -0.3 Part (a) is a routine Simpson's rule application with straightforward function evaluation. Part (b) requires recognizing the standard integral form ∫f'(x)/f(x)dx = ln|f(x)| with u = x² + 2, which is a common C3 technique. The 'show that' format and finding k adds minimal challenge beyond standard integration by substitution. |
| Spec | 1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(h = 1\), ordinates at \(x = 0, 1, 2, 3, 4\) | B1 | Correct strip width |
| \(y_0 = 0,\ y_1 = \frac{1}{3},\ y_2 = \frac{2}{6} = \frac{1}{3},\ y_3 = \frac{3}{11},\ y_4 = \frac{4}{18} = \frac{2}{9}\) | B1 | All ordinates correct |
| \(I \approx \frac{1}{3}[y_0 + 4y_1 + 2y_2 + 4y_3 + y_4]\) | M1 | Correct Simpson's rule structure |
| \(= \frac{1}{3}\left[0 + \frac{4}{3} + \frac{2}{6} + \frac{12}{11} + \frac{2}{9}\right]\) | ||
| \(\approx 0.6870\) | A1 | Answer to 4 significant figures |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^4 \frac{x}{x^2+2}\,dx = \left[\frac{1}{2}\ln(x^2+2)\right]_0^4\) | M1 A1 | Recognise form; correct integration |
| \(= \frac{1}{2}\ln 18 - \frac{1}{2}\ln 2\) | M1 | Apply limits |
| \(= \frac{1}{2}\ln 9 = \ln 3\) | A1 A1 | Simplify to \(\ln k\); \(k = 3\) |
# Question 2:
## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $h = 1$, ordinates at $x = 0, 1, 2, 3, 4$ | B1 | Correct strip width |
| $y_0 = 0,\ y_1 = \frac{1}{3},\ y_2 = \frac{2}{6} = \frac{1}{3},\ y_3 = \frac{3}{11},\ y_4 = \frac{4}{18} = \frac{2}{9}$ | B1 | All ordinates correct |
| $I \approx \frac{1}{3}[y_0 + 4y_1 + 2y_2 + 4y_3 + y_4]$ | M1 | Correct Simpson's rule structure |
| $= \frac{1}{3}\left[0 + \frac{4}{3} + \frac{2}{6} + \frac{12}{11} + \frac{2}{9}\right]$ | | |
| $\approx 0.6870$ | A1 | Answer to 4 significant figures |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^4 \frac{x}{x^2+2}\,dx = \left[\frac{1}{2}\ln(x^2+2)\right]_0^4$ | M1 A1 | Recognise form; correct integration |
| $= \frac{1}{2}\ln 18 - \frac{1}{2}\ln 2$ | M1 | Apply limits |
| $= \frac{1}{2}\ln 9 = \ln 3$ | A1 A1 | Simplify to $\ln k$; $k = 3$ |
---2
\begin{enumerate}[label=(\alph*)]
\item Use Simpson's rule, with five ordinates (four strips), to calculate an estimate for
$$\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x$$
Give your answer to four significant figures.
\item Show that the exact value of $\int _ { 0 } ^ { 4 } \frac { x } { x ^ { 2 } + 2 } \mathrm {~d} x$ is $\ln k$, where $k$ is an integer. (5 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2013 Q2 [9]}}