Challenging +1.8 This is a 10-mark Further Maths integration problem requiring integration by parts with hyperbolic functions. While it involves the non-standard function arcosh(x) and requires knowledge of its derivative and careful algebraic manipulation of surds, the solution follows a predictable pattern: set up the integral, apply integration by parts, and simplify. The main challenges are technical accuracy with hyperbolic identities and surd manipulation rather than novel problem-solving insight.
\includegraphics{figure_33}
Figure 2 shows a sketch of the curve with equation
$$y = x \operatorname{arcosh} x, \quad 1 \leq x \leq 2.$$
The region \(R\), as shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis and the line \(x = 2\).
Show that the area of \(R\) is
$$\frac{7}{4} \ln(2 + \sqrt{3}) - \frac{\sqrt{3}}{2}.$$
(Total 10 marks)
\includegraphics{figure_33}
Figure 2 shows a sketch of the curve with equation
$$y = x \operatorname{arcosh} x, \quad 1 \leq x \leq 2.$$
The region $R$, as shown shaded in Figure 2, is bounded by the curve, the $x$-axis and the line $x = 2$.
Show that the area of $R$ is
$$\frac{7}{4} \ln(2 + \sqrt{3}) - \frac{\sqrt{3}}{2}.$$
(Total 10 marks)
\hfill \mbox{\textit{Edexcel FP3 Q33}}