| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Definite integral with complex substitution requiring algebraic rearrangement |
| Difficulty | Moderate -0.3 Part (a) is routine chain rule application. Part (b) is a standard substitution exercise where the substitution is given explicitly and the algebra is straightforward (expressing x in terms of u, then integrating two simple power terms). This is typical textbook practice for learning the substitution method, slightly easier than average A-level questions which often require more steps or less obvious manipulations. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = (3x-1)^{10}\); \(\frac{dy}{dx} = 10(3x-1)^9 \times 3 = 30(3x-1)^9\) | M1 A1 | M1 for \(a(3x-1)^9\) where \(a\) = constant; 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int x(2x+1)^3 dx\) with \(u = 2x+1\), \(du = 2dx\) | B1 | OE; leads to \(\int \left(\frac{u-1}{2}\right)u^3\left(\frac{du}{2}\right) = \frac{1}{4}\int u^9 - u^8 du\) |
### 2(a)
$y = (3x-1)^{10}$; $\frac{dy}{dx} = 10(3x-1)^9 \times 3 = 30(3x-1)^9$ | M1 A1 | M1 for $a(3x-1)^9$ where $a$ = constant; 2 marks total
### 2(b)
$\int x(2x+1)^3 dx$ with $u = 2x+1$, $du = 2dx$ | B1 | OE; leads to $\int \left(\frac{u-1}{2}\right)u^3\left(\frac{du}{2}\right) = \frac{1}{4}\int u^9 - u^8 du$ | M1 | all in terms of $u$. Condone omission of $du$ | $\frac{1}{4}\left[\frac{u^{10}}{10} + q\frac{u^9}{9}\right]$ | B1 | $p\frac{u^{10}}{10} + q\frac{u^9}{9}$ | $= \frac{(2x+1)^{10}}{40} - \frac{(2x+1)^9}{36} (+c)$ | A1 | OE; CAO; SC: correct answer, no working/parts in $x$ (B1); 4 marks total2
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $y = ( 3 x - 1 ) ^ { 10 }$.
\item Use the substitution $u = 2 x + 1$ to find $\int x ( 2 x + 1 ) ^ { 8 } \mathrm {~d} x$, giving your answer in terms of $x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2006 Q2 [6]}}