32 9. (a) Use a hyperbolic substitution and calculus to show that x x x x x x k 2 2 2 1 1 2 1 − = − +    + ∫ d arcosh where k is an arbitrary constant. (6) y O x = 3 R C x Figure 1 Figure 1 shows a sketch of part of the curve C with equation y = 4 15 x arcosh x x  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 1 15 17 3 2 2 6 2 ln + ( ) −   (5) _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________