Use the substitution \(\mathrm { u } = \mathrm { x } ^ { 2 } - 1\) to find \(\int \frac { x } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x\).
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The diagram shows the curve with equation \(\mathrm { y } = \cosh ^ { - 1 } \mathrm { x }\) together with a set of \(( N - 1 )\) rectangles of unit width.
By considering the sum of the areas of these rectangles, show that
$$\sum _ { r = 2 } ^ { N } \ln \left( r + \sqrt { r ^ { 2 } - 1 } \right) > N \ln \left( N + \sqrt { N ^ { 2 } - 1 } \right) - \sqrt { N ^ { 2 } - 1 }$$
Use a similar method to find, in terms of \(N\), an upper bound for \(\sum _ { \mathrm { r } = 2 } ^ { \mathrm { N } } \ln \left( \mathrm { r } + \sqrt { \mathrm { r } ^ { 2 } - 1 } \right)\).