| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2018 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Integration by Parts |
| Type | Sequential multi-part (building on previous) |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: substitution with careful manipulation in part (a), differentiation of an integral (Fundamental Theorem) in (b)(i), then applying results in (b)(ii), and finally optimizing an integral expression in (c) which requires differentiating under the integral sign. While each individual step uses known techniques, the combination and the non-standard nature of part (c) elevate this significantly above typical A-level questions. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.08a Fundamental theorem of calculus: integration as reverse of differentiation1.08h Integration by substitution |
6. (a) Use the substitution $u = \sqrt { t }$ to show that
$$\int _ { 1 } ^ { x } \frac { \ln t } { \sqrt { t } } \mathrm {~d} t = 4 - 4 \sqrt { x } + 2 \sqrt { x } \ln x \quad x \geqslant 1$$
(b) The function g is such that
$$\int _ { 1 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t = x - \sqrt { x } \ln x - 1 \quad x \geqslant 1$$
\begin{enumerate}[label=(\roman*)]
\item Use differentiation to find the function g .
\item Evaluate $\int _ { 4 } ^ { 16 } \mathrm {~g} ( t ) \mathrm { d } t$ and simplify your answer.\\
(c) Find the value of $x$ (where $x > 1$ ) that gives the maximum value of
$$\int _ { x } ^ { x + 1 } \frac { \ln t } { 2 ^ { t } } \mathrm {~d} t$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2018 Q6 [17]}}