Problem

(a) Use the substitution $x = \frac{1}{2}\sqrt{2}\sinh u$ to find $\int \frac{1}{\sqrt{2x^2 + 1}}\,\mathrm{d}x$.

The diagram shows the curve with equation $y = \frac{1}{\sqrt{2x^2 + 1}}$ for $0 \leq x \leq 1$, together with a set of $n$ rectangles of width $\frac{1}{n}$.

(b) By considering the sum of the areas of these rectangles, show that

$$\sum_{r = 1}^{n}\frac{1}{\sqrt{2r^2 + n^2}} < \frac{1}{2}\sqrt{2}\ln\big(\sqrt{2} + \sqrt{3}\big).$$

(c) Use a similar method to find, in terms of $n$, a lower bound for $\sum_{r = 1}^{n}\frac{1}{\sqrt{2r^2 + n^2}}$.

(d) Deduce the exact value of

$$\lim_{n \to \infty}\sum_{r = 1}^{n} \frac{1}{\sqrt{2r^2 + n^2}}.$$