Problem

The curve $C$ has equation $y = \tanh x$ for $x \ge 0$.

(a) Sketch $C$ and state the equation of the asymptote.

(b) By considering a suitable set of $N$ rectangles of unit width, use your sketch to show that

$$\sum_{r = 1}^{N} \tanh r > \ln(\cosh N).$$

(c) The arc of $C$ joining the point where $x = 0$ to the point where $x = \frac{1}{2}\ln 3$ is rotated through one complete revolution about the $x$-axis. The area of the surface generated is denoted by $S$.

(i) Use the substitution $u = \sqrt{1 + \operatorname{sech}^4 x}$ to show that

$$S = \pi\int_{\frac{5}{4}}^{\sqrt{2}} \frac{u^2}{u^2 - 1}\,\mathrm{d}u.$$

(ii) Find the exact value of

$$\pi\int_{\frac{5}{4}}^{\sqrt{2}} \frac{u^2}{u^2 - 1}\,\mathrm{d}u.$$

You need not simplify your answer.