- (a) Use a hyperbolic substitution and calculus to show that
$$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } \left[ x \sqrt { x ^ { 2 } - 1 } + \operatorname { arcosh } x \right] + k$$
where \(k\) is an arbitrary constant.
\begin{figure}[h]
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\caption{Figure 1}
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Figure 1 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { 4 } { 15 } x \operatorname { arcosh } x \quad x \geqslant 1$$
The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
(b) Using algebraic integration and the result from part (a), show that the area of \(R\) is given by
$$\frac { 1 } { 15 } [ 17 \ln ( 3 + 2 \sqrt { 2 } ) - 6 \sqrt { 2 } ]$$