4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials

1. (a) Show that, under the substitution \(x = \frac { 3 } { 4 } \sinh u\),

$$\int \frac { x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = k \int ( \cosh 2 u - 1 ) \mathrm { d } u$$ where \(k\) is a constant to be determined.
(b) Hence show that $$\int _ { 0 } ^ { 1 } \frac { 64 x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = p + q \ln 3$$ where \(p\) and \(q\) are rational numbers to be found.

1. The curve \(C\) has equation

$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.

8. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 1\)
2. Hence show that $$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$ where \(p\) is a rational number to be found.
   8. \(\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0\)
   1. Show that, for \(n \geqslant 1\) $$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$

1. Solve the equation

$$18 \cosh x + 14 \sinh x = 11 + \mathrm { e } ^ { x }$$ Give your answers in the form \(\ln a\), where \(a\) is rational.

1. Solve the equation

$$15 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 13$$ Give your answers in terms of simplified natural logarithms.

6. The curve \(C\) has parametric equations $$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$

1. Find
   1. \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\)
   2. \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\) The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the exact area of the curved surface formed, giving your answer as a multiple of \(\pi\).

1. Find the exact values of x for which

$$4 \tanh ^ { 2 } x - 2 \operatorname { sech } ^ { 2 } x = 3 ,$$ giving your answers in the form \(\pm \ln \mathrm { a }\), where a is real.

2. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = 2 \cosh \left( \frac { 1 } { 2 } x \right)\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates \(- \ln 2\) and \(\ln 2\) respectively. The arc of the curve joining \(A\) and \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the exact area of the curved surface area formed.
(Total 7 marks)

1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that

$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.

1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that

$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve the equation

$$7 \cosh x + 3 \sinh x = 2 \mathrm { e } ^ { x } + 7$$ Give your answers as simplified natural logarithms.

7. $$I _ { n } = \int \cosh ^ { n } 2 x \mathrm {~d} x \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { \cosh ^ { n - 1 } 2 x \sinh 2 x } { 2 n } + \frac { n - 1 } { n } I _ { n - 2 }$$
2. Hence determine $$\int ( 1 + \cosh 2 x ) ^ { 3 } d x$$ collecting any like terms in your answer.

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

$$\sinh ( A + B ) \equiv \sinh A \cosh B + \cosh A \sinh B$$ (b) Hence express \(10 \sinh x + 8 \cosh x\) in the form \(R \sinh ( x + \alpha )\) where \(R > 0\), giving \(\alpha\) in the form \(\ln p\) where \(p\) is an integer.
(c) Hence solve the equation $$10 \sinh x + 8 \cosh x = 18 \sqrt { 7 }$$ giving your answer in the form \(\ln ( \sqrt { 7 } + q )\) where \(q\) is a rational number to be determined.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows the curve with equation $$y = \ln \left( \tanh \frac { x } { 2 } \right) \quad 1 \leqslant x \leqslant 2$$

1. Show that the length, \(s\), of the curve is given by $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
2. Hence show that $$s = \ln \left( \mathrm { e } + \frac { 1 } { \mathrm { e } } \right)$$

2. Given that $$\cosh y = x \quad \text { and } \quad y < 0$$ use the definition of coshy in terms of exponential functions to prove that $$y = \ln \left( x - \sqrt { x ^ { 2 } - 1 } \right)$$

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\) where \(a\) is a rational number.

3. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ (b) Solve the equation $$\cosh 2 x - 3 \sinh x = 15$$ giving your answers as exact logarithms.

2. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has equation $$y = \frac { 1 } { 3 } \cosh 3 x , \quad 0 \leqslant x \leqslant \ln a$$ where \(a\) is a constant and \(a > 1\) Using calculus, show that the length of curve \(C\) is $$k \left( a ^ { 3 } - \frac { 1 } { a ^ { 3 } } \right)$$ and state the value of the constant \(k\).

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 \(\int _ { \frac { 1 } { 2 } \ln 3 } ^ { \ln 3 } \frac { 1 } { 5 \cosh x - 4 \sinh x } \mathrm {~d} x = \frac { \pi } { 18 }\)

7. \begin{figure}[h] \end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.

3. The curve with parametric equations $$x = \cosh 2 \theta , \quad y = 4 \sinh \theta , \quad 0 \leqslant \theta \leqslant 1$$ is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the surface generated is \(\lambda \left( \cosh ^ { 3 } \alpha - 1 \right)\), where \(\alpha = 1\) and \(\lambda\) is a constant to be found.

1. Solve the equation

$$5 \tanh x + 7 = 5 \operatorname { sech } x$$ Give each answer in the form \(\ln k\) where \(k\) is a rational number.

1. The curve \(C\) has equation

$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$ The points \(A\) and \(B\) lie on \(C\). The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
2. Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
   (6)

1. Solve the equation

$$2 \cosh ^ { 2 } x - 3 \sinh x = 1$$ giving your answers in terms of natural logarithms.

2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .