Express hyperbolic in exponential form

4 Showing all necessary working, solve the equation $$\frac { \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } } { \mathrm { e } ^ { x } + 1 } = 4$$ giving your answer correct to 3 decimal places.

7. $$\mathrm { f } ( x ) = 5 \cosh x - 4 \sinh x , \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + 9 \mathrm { e } ^ { - x } \right)\) Hence
2. solve \(\mathrm { f } ( x ) = 5\)
3. show that \(\int _ { \frac { 1 } { 2 } \ln 3 } ^ { \ln 3 } \frac { 1 } { 5 \cosh x - 4 \sinh x } \mathrm {~d} x = \frac { \pi } { 18 }\)

1 Express sech \(2 x\) in terms of exponentials and hence, by using the substitution \(u = e ^ { 2 x }\), find \(\int \operatorname { sech } 2 x \mathrm {~d} x\).

1 Which expression below is equivalent to \(\tanh x\) ? Circle your answer.
\(\sinh x \cosh x\)
\(\frac { \sinh x } { \cosh x }\)
\(\frac { \cosh x } { \sinh x }\)
\(\sinh x + \cosh x\)

12 The function \(\mathrm { f } ( x ) = \cosh ( \mathrm { i } x )\) is defined over the domain \(\{ x \in \mathbb { R } : - a \pi \leq x \leq a \pi \}\), where \(a\) is a positive integer. By considering the graph of \(y = [ f ( x ) ] ^ { n }\), find the mean value of \([ f ( x ) ] ^ { n }\), when \(n\) is an odd positive integer. Fully justify your answer.
[0pt] [3 marks]

7 The function sech \(x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).

1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).

1 Which of the following exponential expressions is equivalent to \(2 \sinh x\) ?
Circle your answer.
\(\mathrm { e } ^ { x }\)
\(\mathrm { e } ^ { x } + \mathrm { e } ^ { - x }\)
\(\mathrm { e } ^ { x } - \mathrm { e } ^ { - x }\)
\(\mathrm { e } ^ { - x }\)

6

1. Show that $$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$ where \(m\) and \(n\) are integers.
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2. Hence find \(\cosh ^ { 6 } x - \sinh ^ { 6 } x\) in the form $$\frac { a \cosh ( k x ) + b } { 8 }$$ where \(a , b\) and \(k\) are integers.

5 The function \(\operatorname { sech } x\) is defined by \(\operatorname { sech } x = \frac { 1 } { \cosh x }\).

1. Show that \(\operatorname { sech } x = \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } + 1 }\).
2. Using a suitable substitution, find \(\int \operatorname { sech } x \mathrm {~d} x\).