| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Substitution then integration by parts |
| Difficulty | Standard +0.3 This is a structured multi-part question that guides students through substitution followed by integration by parts. Part (a) is standard integration by parts with ln t. Part (b) is a straightforward substitution verification. Part (c) applies the result to definite integration. The scaffolding makes this slightly easier than average, though it requires careful execution of multiple techniques. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
7
\begin{enumerate}[label=(\alph*)]
\item Use integration by parts to find $\int ( t - 1 ) \ln t \mathrm {~d} t$.
\item Use the substitution $t = 2 x + 1$ to show that $\int 4 x \ln ( 2 x + 1 ) \mathrm { d } x$ can be written as $\int ( t - 1 ) \ln t \mathrm {~d} t$.
\item Hence find the exact value of $\int _ { 0 } ^ { 1 } 4 x \ln ( 2 x + 1 ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2009 Q7 [10]}}