Differentiate inverse hyperbolic functions

4. $$y = \operatorname { artanh } \left( \frac { \cos x + a } { \cos x - a } \right)$$ where \(a\) is a non-zero constant.
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \tan x$$ where \(k\) is a constant to be determined.

1. (a) Differentiate \(x \operatorname { arcosh } 5 x\) with respect to \(x\)
   (b) Hence, or otherwise, show that

$$\int _ { \frac { 1 } { 4 } } ^ { \frac { 3 } { 5 } } \operatorname { arcosh } 5 x \mathrm {~d} x = \frac { 3 } { 20 } - \frac { 2 \sqrt { 2 } } { 5 } + \ln ( p + q \sqrt { 2 } ) ^ { k } - \frac { 1 } { 4 } \ln r$$ where \(p , q , r\) and \(k\) are rational numbers to be determined.

1. (a) Differentiate \(x \operatorname { arsinh } 2 x\) with respect to \(x\).
   (b) Hence, or otherwise, find the exact value of

$$\int _ { 0 } ^ { \sqrt { 2 } } \operatorname { arsinh } 2 x \mathrm {~d} x$$ giving your answer in the form \(A \ln B + C\), where \(A , B\) and \(C\) are real.

1. Given that \(y = \operatorname { arsinh } ( \tanh x )\), show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \operatorname { sech } ^ { 2 } x } { \sqrt { 1 + \tanh ^ { 2 } x } }$$ \section*{-} \includegraphics[max width=\textwidth, alt={}, center]{64dc962a-1788-49ac-a4db-af1241b552a0-03_51_51_276_2012}
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4

1. Solve the equation $$\sinh x + 4 \cosh x = 8$$ giving the answers in an exact logarithmic form.
2. Find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x } \sinh x \mathrm {~d} x\).
3. 1. Differentiate \(\operatorname { arsinh } \left( \frac { 2 } { 3 } x \right)\) with respect to \(x\).
   2. Use integration by parts to show that \(\int _ { 0 } ^ { 2 } \operatorname { arsinh } \left( \frac { 2 } { 3 } x \right) \mathrm { d } x = 2 \ln 3 - 1\).

3

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }\).
2. Find the equation of the normal to the graph of \(\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }\) at the point where \(x = \sqrt { 6 }\). Give your answer in the form \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) where \(m\) and \(c\) are given in exact, non-hyperbolic form.

13

1. Using the logarithmic form of \(\operatorname { arcosh } x\), prove that the derivative of \(\operatorname { arcosh } x\) is \(\frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\).
2. Hence find \(\int _ { 1 } ^ { 2 } \operatorname { arcosh } x \mathrm {~d} x\), giving your answer in exact logarithmic form.
3. Ali tries to evaluate \(\int _ { 0 } ^ { 1 } \operatorname { arcosh } x \mathrm {~d} x\) using his calculator, and gets an 'error'. Explain why.

9. (a) Given that \(y = \sin ^ { - 1 } ( \cos \theta )\), where \(0 \leqslant \theta \leqslant \pi\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} \theta } = k\), where the value of \(k\) is to be determined.
(b) Find the value of the gradient of the curve \(y = x ^ { 3 } \tan ^ { - 1 } 4 x\) when \(x = \frac { \pi } { 2 }\).
(c) Find the equation of the normal to the curve \(y = \tanh ^ { - 1 } ( 1 - x )\) when \(x = 1 \cdot 7\).

15 In this question you must show detailed reasoning. Show that $$\int _ { 0 } ^ { \frac { 2 } { 3 } } \operatorname { arsinh } 2 x \mathrm {~d} x = \frac { 2 } { 3 } \ln 3 - \frac { 1 } { 3 }$$

14. The curve \(C\) has equation $$y = \operatorname { arcsec } \mathrm { e } ^ { x } , \quad x > 0 , \quad 0 < y < \frac { 1 } { 2 } \pi$$

1. Prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( \mathrm { e } ^ { 2 x } - 1 \right) } }\).
2. Sketch the graph of \(C\). The point \(A\) on \(C\) has \(x\)-coordinate \(\ln 2\). The tangent to \(C\) at \(A\) intersects the \(y\)-axis at the point \(B\).
3. Find the exact value of the \(y\)-coordinate of \(B\).