| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Show definite integral equals value |
| Difficulty | Standard +0.3 Part (a) is a straightforward application of Simpson's rule with clear ordinates requiring only careful arithmetic. Part (b) requires recognizing that the derivative of 1+x³ is 3x², leading to a standard logarithmic integral—a common C3 technique but slightly above routine since it requires pattern recognition for the substitution. |
| Spec | 1.08h Integration by substitution1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(h = \frac{2-0.5}{6} = 0.25\); ordinates at \(x = 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2\) | M1 | |
| Simpson's rule: \(\frac{0.25}{3}[y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6]\) | M1 | |
| \(\approx 0.511\) | A2 | A1 for correct structure, A1 for answer |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int_0^1 \frac{x^2}{1+x^3}dx\); let \(u = 1+x^3\), \(du = 3x^2\,dx\) | M1 | |
| \(= \frac{1}{3}\int_1^2 \frac{1}{u}du = \frac{1}{3}[\ln u]_1^2\) | M1 A1 | |
| \(= \frac{1}{3}\ln 2\) | A1 | Exact value required |
# Question 4:
## Part (a)
| $h = \frac{2-0.5}{6} = 0.25$; ordinates at $x = 0.5, 0.75, 1, 1.25, 1.5, 1.75, 2$ | M1 | |
| Simpson's rule: $\frac{0.25}{3}[y_0 + 4y_1 + 2y_2 + 4y_3 + 2y_4 + 4y_5 + y_6]$ | M1 | |
| $\approx 0.511$ | A2 | A1 for correct structure, A1 for answer |
## Part (b)
| $\int_0^1 \frac{x^2}{1+x^3}dx$; let $u = 1+x^3$, $du = 3x^2\,dx$ | M1 | |
| $= \frac{1}{3}\int_1^2 \frac{1}{u}du = \frac{1}{3}[\ln u]_1^2$ | M1 A1 | |
| $= \frac{1}{3}\ln 2$ | A1 | Exact value required |
---4
\begin{enumerate}[label=(\alph*)]
\item Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $\int _ { 0.5 } ^ { 2 } \frac { x } { 1 + x ^ { 3 } } \mathrm {~d} x$, giving your answer to three significant figures.
\item Find the exact value of $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { 1 + x ^ { 3 } } \mathrm {~d} x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2010 Q4 [8]}}