| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Independent multi-part (different techniques) |
| Difficulty | Moderate -0.3 Part (a) is a textbook integration by parts with polynomial×exponential. Parts (b)(i-ii) involve a straightforward substitution u=√x followed by standard logarithm integration. All techniques are routine C3 applications with no problem-solving insight required, making this slightly easier than average but not trivial due to the multi-step nature and exact value requirement. |
| Spec | 1.08h Integration by substitution1.08i Integration by parts |
6
\begin{enumerate}[label=(\alph*)]
\item Use integration by parts to find $\int x \mathrm { e } ^ { 5 x } \mathrm {~d} x$.
\item \begin{enumerate}[label=(\roman*)]
\item Use the substitution $u = \sqrt { x }$ to show that
$$\int \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x = \int \frac { 2 } { 1 + u } \mathrm {~d} u$$
\item Find the exact value of $\int _ { 1 } ^ { 9 } \frac { 1 } { \sqrt { x } ( 1 + \sqrt { x } ) } \mathrm { d } x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q6 [9]}}