Hyperbolic functions：練習問題

Find the exact value of \(\cosh(\ln 4)\), giving your answer as a simplified fraction.

Find the exact value of \(\text{arsinh}\left(\frac{3}{4}\right)\), giving your answer in logarithmic form.

Solve the equation \(5\cosh x - \sinh x = 7\) for all real values of \(x\). Give your answer in exact logarithmic form.

Using the exponential definitions of hyperbolic functions, find the exact value of \(\sinh(\ln 3)\).

Given that \(y = \text{artanh}(\sin x)\) for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), find an expression for \(\frac{dy}{dx}\) in its simplest form.

Calculate the exact value of \(\sinh(\ln 3)\), giving your answer as a fraction.

Find the gradient of the curve \(y = \text{arsinh}(x^2)\) at the point where \(x = 1\), giving your answer in simplest surd form.<\/p>

Find the exact value of the integral \(\int_{1}^{2} \frac{1}{\sqrt{x^2 + 4x}} \, dx\), giving your answer in the form \(\ln \left( \frac{a + \sqrt{b}}{c + \sqrt{d}} \right)\) where \(a, b, c, d\) are integers.<\/p>

Find the exact mean value of the function \(f(x) = \sinh 2x\) over the interval \([0, \ln 2]\). Give your answer in the form \(\frac{a}{b \ln c}\), where \(a, b, c\) are integers.
Note: the mean value of a function over the interval \([a, b]\) is given by \(\frac{1}{b-a}\int_{a}^{b} f(x) dx\).

(a) Given that \(y = \text{artanh } x\), where \(|x| < 1\), show that \(\frac{dy}{dx} = \frac{1}{1-x^2}\).
(b) Using integration by parts, show that \(\int \text{artanh } x \, dx = x \text{artanh } x + \frac{1}{2} \ln(1-x^2) + C\).
(c) Hence find the exact value of the area enclosed by the curve \(y = \text{artanh } x\), the \(x\)-axis, and the line \(x = \frac{1}{2}\). Give your answer in the form \(\frac{1}{a} \ln b - \ln c\) where \(a, b, c\) are positive integers.