| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2022 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integration with substitution given |
| Difficulty | Challenging +1.2 This is a multi-part integration question requiring differentiation of a composite exponential, solving a polynomial equation, and integration using reverse chain rule. While it involves several steps and the exponential composite function, the techniques are standard A-level methods (product rule, reverse chain rule recognition) with straightforward algebra. The substitution structure is transparent once f'(x) is found, making this moderately above average but not requiring novel insight. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08e Area between curve and x-axis: using definite integrals |
4.Given that $\mathrm { f } ( x ) = \mathrm { e } ^ { x ^ { 3 } - 2 x }$
\begin{enumerate}[label=(\alph*)]
\item find $\mathrm { f } ^ { \prime } ( x )$
The curves $C _ { 1 }$ and $C _ { 2 }$ are defined by the functions g and h respectively,where
$$\begin{array} { l l }
\mathrm { g } ( x ) = 8 x ^ { 3 } \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0 \\
\mathrm {~h} ( x ) = \left( 3 x ^ { 5 } + 4 x \right) \mathrm { e } ^ { x ^ { 3 } - 2 x } & x \in \mathbb { R } , x > 0
\end{array}$$
\item Find the $x$ coordinates of the points of intersection of $C _ { 1 }$ and $C _ { 2 }$
Given that $C _ { 1 }$ lies above $C _ { 2 }$ between these points of intersection,
\item find the area of the region bounded by the curves between these two points.\\
Give your answer in the form $A + B \mathrm { e } ^ { C }$ where $A , B$ ,and $C$ are exact real numbers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2022 Q4 [14]}}