Reduction formulas with hyperbolic integrals

Edexcel F3 2024 January Q5

11 marks Challenging +1.3

(a) Use the definitions of hyperbolic functions in terms of exponentials to prove that

$$\begin{gathered} 1 - \operatorname { sech } ^ { 2 } x \equiv \tanh ^ { 2 } x \\ I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { n } 3 x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0 \end{gathered}$$ (b) Show that $$I _ { n } = I _ { n - 2 } - \frac { p ^ { n - 1 } } { 3 ( n - 1 ) } \quad n \geqslant 2$$ where $p$ is a rational number to be determined.
(c) Hence determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { 5 } 3 x \mathrm {~d} x$$ giving your answer in the form $a \ln b + c$ where $a , b$ and $c$ are rational numbers to be found.

OCR FP2 2011 January Q8

12 marks Challenging +1.8

8

Without using a calculator, show that $\sinh \left( \cosh ^ { - 1 } 2 \right) = \sqrt { 3 }$.
It is given that, for non-negative integers $n$, $$I _ { n } = \int _ { 0 } ^ { \beta } \cosh ^ { n } x \mathrm {~d} x , \quad \text { where } \beta = \cosh ^ { - 1 } 2$$ Show that $n I _ { n } = 2 ^ { n - 1 } \sqrt { 3 } + ( n - 1 ) I _ { n - 2 }$, for $n \geqslant 2$.
Evaluate $I _ { 5 }$, giving your answer in the form $k \sqrt { 3 }$.

Pre-U Pre-U 9795/1 2013 June Q13

4 marks Challenging +1.8

13

Use the definitions $\tanh \theta = \frac { \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }$ and $\operatorname { sech } \theta = \frac { 2 } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }$ to prove the results
1. $\tanh ^ { 2 } \theta \equiv 1 - \operatorname { sech } ^ { 2 } \theta$,
2. $\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tanh \theta ) = \operatorname { sech } ^ { 2 } \theta$.
3. Let $I _ { n } = \int _ { 0 } ^ { \alpha } \tanh ^ { 2 n } \theta \mathrm {~d} \theta$ for $n \geqslant 0$, where $\alpha > 0$.
  (a) Show that $I _ { n - 1 } - I _ { n } = \frac { \tanh ^ { 2 n - 1 } \alpha } { 2 n - 1 }$ for $n \geqslant 1$. Given that $\alpha = \frac { 1 } { 2 } \ln 3$,
  (b) evaluate $I _ { 0 }$,
4. use the method of differences to show that $I _ { n } = \frac { 1 } { 2 } \ln 3 - \sum _ { r = 1 } ^ { n } \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 r - 1 } } { 2 r - 1 }$ and deduce the sum of the infinite series $\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }$.

CAIE Further Paper 2 2021 November Q8

13 marks Challenging +1.2

Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh^2 x = \sech^2 x.$$ [3]
Using the substitution $u = \tanh x$, or otherwise, find $\int \sech^2 x \tanh^2 x \, dx$. [2]
It is given that, for $n \geq 0$, $I_n = \int_0^{\ln 3} \sech^n x \tanh^2 x \, dx$. Show that, for $n \geq 2$, $$(n + 1)I_n = \left(\frac{4}{3}\right)^{\frac{3}{n-2}} + (n - 2)I_{n-2}.$$ [You may use the result that $\frac{d}{dx}(\sech x) = -\tanh x \sech x$.] [5]
Find the value of $I_4$. [3]

Edexcel F3 2018 Specimen Q8

10 marks Challenging +1.8

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

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

SPS SPS FM 2021 November Q6

7 marks Challenging +1.8

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]

Pre-U Pre-U 9795/1 2013 November Q13

24 marks Hard +2.3

Let $I_n = \int_0^{\alpha} \cosh^n x \, dx$ for integers $n \geqslant 0$, where $\alpha = \ln 2$.
1. Prove that, for $n \geqslant 2$, $nI_n = \frac{3 \times 5^{n-1}}{4^n} + (n-1)I_{n-2}$. [5]
2. A curve has parametric equations $x = 12 \sinh t + 4 \sinh^3 t$, $y = 3 \cosh^4 t$, $0 \leqslant t \leqslant \ln 2$. Find the length of the arc of this curve, giving your answer in the form $a + b \ln 2$ for rational numbers $a$ and $b$. [8]
The circle with equation $x^2 + (y - R)^2 = r^2$, where $r < R$, is rotated through one revolution about the $x$-axis to form a solid of revolution called a torus. By using suitable parametric equations for the circle, determine, in terms of $\pi$, $R$ and $r$, the surface area of this torus. [11]