| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2019 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Derivative then integrate by parts |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: finding where sin(ln x) = 0, using product rule with chain rule on composite functions, recognizing a system of integrals that must be solved simultaneously, and applying integration by parts to x·sin(ln x). The guided structure helps, but the composite function differentiation and the coupled integral system require significant mathematical maturity beyond standard A-level. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08e Area between curve and x-axis: using definite integrals1.08i Integration by parts |
6.Figure 1 shows a sketch of part of the curve with equation $y = x \sin ( \ln x ) , x \geqslant 1$
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{175528b0-6cd1-4d0d-a6b3-28ac980f74f3-18_451_1170_312_450}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
For $x > 1$ ,the curve first crosses the $x$-axis at the point $A$ .
\begin{enumerate}[label=(\alph*)]
\item Find the $x$ coordinate of $A$ .
\item Differentiate $x \sin ( \ln x )$ and $x \cos ( \ln x )$ with respect to $x$ and hence find
$$\int \sin ( \ln x ) \mathrm { d } x \text { and } \int \cos ( \ln x ) \mathrm { d } x$$
\item \begin{enumerate}[label=(\roman*)]
\item Find $\int x \sin ( \ln x ) \mathrm { d } x$ .
\item Hence show that the area of the shaded region $\boldsymbol { R }$ ,bounded by the curve and the $x$-axis between the points $( 1,0 )$ and $A$ ,is
$$\frac { 1 } { 5 } \left( \mathrm { e } ^ { 2 \pi } + 1 \right)$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2019 Q6 [19]}}