| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Square root substitution: definite integral |
| Difficulty | Standard +0.3 Part (a) is a standard integration by parts exercise testing routine application of the formula with ln x, a common textbook example. Part (b) requires the substitution u = √x (which is given), then simplifying the resulting integral—this is a typical C3/C4 substitution question with straightforward algebraic manipulation. The 7 marks for part (b) reflect multiple steps rather than conceptual difficulty. Overall slightly easier than average due to the substitution being provided and both techniques being standard applications. |
| Spec | 1.08i Integration by parts |
10
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item By writing $\ln x$ as $( \ln x ) \times 1$, use integration by parts to find $\int \ln x \mathrm {~d} x$.
\item Find $\int ( \ln x ) ^ { 2 } \mathrm {~d} x$.\\
(4 marks)
\end{enumerate}\item Use the substitution $u = \sqrt { x }$ to find the exact value of
$$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$
(7 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2013 Q10 [15]}}