Express hyperbolic in exponential form

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. \(\sinh x \cosh x\) \(\frac { \sinh x } { \cosh x }\) \(\frac { \cosh x } { \sinh x }\) \(\sinh x + \cosh x\)

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. \(\mathrm { e } ^ { x }\) \(\mathrm { e } ^ { x } + \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { x } - \mathrm { e } ^ { - x }\) \(\mathrm { e } ^ { - x }\)

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 \(\int \operatorname { sech } x \mathrm {~d} x\).

Evaluate the integrals

1. \(\int_{-2}^{0} e^{2x} \sinh x \, \mathrm{d}x\), [5]
2. \(\int_{\frac{1}{2}}^{3} \frac{5}{(x-1)(x^2+9)} \, \mathrm{d}x\). [9]