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

$$I_n = \int_{0}^{\ln 2} \tanh^{2n} x \, dx, \quad n \geq 0$$

1. Show that, for \(n \geq 1\) $$I_n = I_{n-1} - \frac{1}{2n-1}\left(\frac{3}{5}\right)^{2n-1}$$ [5]
2. Hence show that $$\int_{0}^{\ln 2} \tanh^{-1} x \, dx = p + \ln 2$$ where \(p\) is a rational number to be found. [5]

The curve \(C_1\) has equation \(y = 3\sinh 2x\), and the curve \(C_2\) has equation \(y = 13 - 3e^{2x}\).

1. Sketch the graph of the curves \(C_1\) and \(C_2\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes. [4]
2. Solve the equation \(3\sinh 2x = 13 - 3e^{2x}\), giving your answer in the form \(\frac{1}{2}\ln k\), where \(k\) is an integer. [5]

The hyperbola \(H\) has equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

1. Use calculus to show that the equation of the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\) may be written in the form $$xb\cosh\theta - ya\sinh\theta = ab$$ [4] The line \(l_1\) is the tangent to \(H\) at the point \((a\cosh\theta, b\sinh\theta)\), \(\theta \neq 0\). Given that \(l_1\) meets the \(x\)-axis at the point \(P\),
2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). [2] The line \(l_2\) is the tangent to \(H\) at the point \((a, 0)\). Given that \(l_1\) and \(l_2\) meet at the point \(Q\),
3. find, in terms of \(a\), \(b\) and \(\theta\), the coordinates of \(Q\). [2]
4. Show that, as \(\theta\) varies, the locus of the mid-point of \(PQ\) has equation $$x(4y^2 + b^2) = ab^2$$ [6]

Using the definitions of hyperbolic functions in terms of exponentials,

1. show that $$\operatorname{sech}^2 x = 1 - \tanh^2 x$$ [3]
2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$ [4]

Solve the equation $$10 \cosh x + 2 \sinh x = 11.$$ Give each answer in the form \(\ln a\) where \(a\) is a rational number. [7]

\includegraphics{figure_6} The curve \(C\) shown in Fig. 1 has equation \(y^2 = 4x\), \(0 \leq x \leq 1\). The part of the curve in the first quadrant is rotated through \(2\pi\) radians about the \(x\)-axis.

1. Show that the surface area of the solid generated is given by $$4\pi \int_0^1 \sqrt{1+x} \, dx.$$ [4]
2. Find the exact value of this surface area. [3]
3. Show also that the length of the curve \(C\), between the points \((1, -2)\) and \((1, 2)\), is given by $$2 \int_0^1 \sqrt{\frac{x+1}{x}} \, dx.$$ [3]
4. Use the substitution \(x = \sinh^2 \theta\) to show that the exact value of this length is $$2[\sqrt{2} + \ln(1 + \sqrt{2})].$$ [6]

Prove that \(\sinh(i\pi - \theta) = \sinh \theta\). [4]

Find the values of \(x\) for which $$4 \cosh x + \sinh x = 8,$$ giving your answer as natural logarithms. [6]

\includegraphics{figure_13} A rope is hung from points \(P\) and \(Q\) on the same horizontal level, as shown in Fig. 2. The curve formed by the rope is modelled by the equation $$y = a \cosh\left(\frac{x}{a}\right), \quad -ka \leq x \leq ka,$$ where \(a\) and \(k\) are positive constants.

1. Prove that the length of the rope is \(2a \sinh k\). [5]

Given that the length of the rope is \(8a\),

1. find the coordinates of \(Q\), leaving your answer in terms of natural logarithms and surds, where appropriate. [4]

$$I_n = \int_0^1 x^n e^x \, dx \text{ and } J_n = \int_0^1 x^n e^{-x} \, dx, \quad n \geq 0.$$

1. Show that, for \(n \geq 1\), $$I_n = e - nI_{n-1}.$$ [2]
2. Find a similar reduction formula for \(J_n\). [3]
3. Show that \(J_2 = 2 - \frac{5}{e}\). [3]
4. Show that \(\int_0^1 x^n \cosh x \, dx = \frac{1}{2}(I_n + J_n)\). [1]
5. Hence, or otherwise, evaluate \(\int_0^1 x^2 \cosh x \, dx\), giving your answer in terms of \(e\). [4]

Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of exponentials,

1. prove that \(\cosh^2 x - \sinh^2 x = 1\), [3]
2. solve \(\operatorname{cosech} x - 2 \coth x = 2\), giving your answer in the form \(k \ln a\), where \(k\) and \(a\) are integers. [4]

1. Show that, for \(x = \ln k\), where \(k\) is a positive constant, $$\cosh 2x = \frac{k^4 + 1}{2k^2}.$$ [3]

Given that \(f(x) = px - \tanh 2x\), where \(p\) is a constant,

1. find the value of \(p\) for which \(f(x)\) has a stationary value at \(x = \ln 2\), giving your answer as an exact fraction. [4]

(Total 7 marks)

1. Show that, for \(0 < x \leq 1\), $$\ln \left(\frac{1 - \sqrt{1-x^2}}{x}\right) = -\ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [3]
2. Using the definition of \(\cosh x\) or \(\operatorname{sech} x\) in terms of exponentials, show that, for \(0 < x \leq 1\), $$\operatorname{arsech} x = \ln \left(\frac{1 + \sqrt{1-x^2}}{x}\right).$$ [5]
3. Solve the equation $$3 \tanh^2 x - 4 \operatorname{sech} x + 1 = 0,$$ giving exact answers in terms of natural logarithms. [5]

(Total 13 marks)

Find the values of \(x\) for which $$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms. (Total 6 marks)

\includegraphics{figure_3} A curve with no stationary points has equation \(y = f(x)\). The equation \(f(x) = 0\) has one real root \(\alpha\), and the Newton-Raphson method is to be used to find \(\alpha\). The tangent to the curve at the point \((x_1, f(x_1))\) meets the \(x\)-axis where \(x = x_2\) (see diagram).

1. Show that \(x_2 = x_1 - \frac{f(x_1)}{f'(x_1)}\). [3]
2. Describe briefly, with the help of a sketch, how the Newton-Raphson method, using an initial approximation \(x = x_1\), gives a sequence of approximations approaching \(\alpha\). [2]
3. Use the Newton-Raphson method, with a first approximation of 1, to find a second approximation to the root of \(x^2 - 2\sinh x + 2 = 0\). [2]

1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]

It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).

1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]

1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]

A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).

1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]

It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).

1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]

Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]