Reduction formulas with hyperbolic integrals

8

1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } x = \operatorname { sech } ^ { 2 } x$$
2. Using the substitution \(\mathrm { u } = \tanh \mathrm { x }\), or otherwise, find \(\int \operatorname { sech } ^ { 2 } x \tanh ^ { 2 } x \mathrm {~d} x\).
   It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } x \tanh ^ { 2 } x \mathrm { dx }\).
3. Show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = \left( \frac { 4 } { 5 } \right) ^ { 3 } \left( \frac { 3 } { 5 } \right) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
4. Find the value of \(I _ { 4 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1. (a) Use the definitions of hyperbolic functions in terms of exponentials to prove that

$$\begin{gathered} 1 - \operatorname { sech } ^ { 2 } x \equiv \tanh ^ { 2 } x \\ I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { n } 3 x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0 \end{gathered}$$ (b) Show that $$I _ { n } = I _ { n - 2 } - \frac { p ^ { n - 1 } } { 3 ( n - 1 ) } \quad n \geqslant 2$$ where \(p\) is a rational number to be determined.
(c) Hence determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { 5 } 3 x \mathrm {~d} x$$ giving your answer in the form \(a \ln b + c\) where \(a , b\) and \(c\) are rational numbers to be found.

8

1. Without using a calculator, show that \(\sinh \left( \cosh ^ { - 1 } 2 \right) = \sqrt { 3 }\).
2. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \beta } \cosh ^ { n } x \mathrm {~d} x , \quad \text { where } \beta = \cosh ^ { - 1 } 2$$ Show that \(n I _ { n } = 2 ^ { n - 1 } \sqrt { 3 } + ( n - 1 ) I _ { n - 2 }\), for \(n \geqslant 2\).
3. Evaluate \(I _ { 5 }\), giving your answer in the form \(k \sqrt { 3 }\).