Standard +0.3 This is a straightforward integration question requiring factorization of the integrand, finding the x-intercept, and applying integration by parts once. The algebraic manipulation is routine and the techniques are standard A-level Further Maths content, making it slightly easier than average.
\includegraphics{figure_4}
Figure 4 shows a sketch of part of the curve with equation
$$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.
Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[5]
\includegraphics{figure_4}
Figure 4 shows a sketch of part of the curve with equation
$$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$
The finite region $R$, shown shaded in Figure 4, is bounded by the curve, the $x$-axis and the $y$-axis.
Use calculus to show that the exact area of $R$ can be written in the form $pe^t + q$, where $p$ and $q$ are rational constants to be found.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
[5]
\hfill \mbox{\textit{SPS SPS FM 2020 Q7 [5]}}