Hyperbolic function reduction

A question is this type if and only if the integral I_n involves hyperbolic functions (sinh, cosh, tanh, sech) and requires deriving or using a reduction formula.

6 questions · Challenging +1.8

4.07a Hyperbolic definitions: sinh, cosh, tanh as exponentials
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CAIE Further Paper 2 2024 June Q4
8 marks Challenging +1.8
4 It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } \mathrm { xdx }\).
  1. Show that, for \(n \geqslant 2\), $$( n - 1 ) \mathrm { I } _ { n } = \left( \frac { 3 } { 5 } \right) ^ { n - 2 } \left( \frac { 4 } { 5 } \right) + ( n - 2 ) \mathrm { I } _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { dx } } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
  2. Find the value of \(I _ { 4 }\).
Edexcel F3 2015 June Q4
10 marks Challenging +1.8
4. $$I _ { n } = \int \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\) $$n I _ { n } = \sinh x \cosh ^ { n - 1 } x + ( n - 1 ) I _ { n - 2 }$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 5 } x \mathrm {~d} x$$
Edexcel F3 2023 June Q7
9 marks Challenging +1.8
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.
Edexcel FP3 2017 June Q7
10 marks Challenging +1.8
7. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 3 a ^ { n - 1 } } { n b ^ { n } } + \frac { n - 1 } { n } I _ { n - 2 }$$ where \(a\) and \(b\) are integers to be found.
  2. Hence, or otherwise, find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 4 } x \mathrm {~d} x$$
OCR FP2 2006 June Q9
13 marks Challenging +1.8
9
  1. Given that \(y = \sinh ^ { - 1 } x\), prove that \(y = \ln \left( x + \sqrt { x ^ { 2 } + 1 } \right)\).
  2. It is given that, for non-negative integers \(n\), $$I _ { n } = \int _ { 0 } ^ { \alpha } \sinh ^ { n } \theta \mathrm {~d} \theta$$ where \(\alpha = \sinh ^ { - 1 } 1\). Show that $$n I _ { n } = \sqrt { 2 } - ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2 .$$
  3. Evaluate \(I _ { 4 }\), giving your answer in terms of \(\sqrt { 2 }\) and logarithms.
Edexcel FP3 Q24
9 marks Challenging +1.8
Given that \(y = \sinh^{n-1} x \cosh x\),
  1. show that \(\frac{dy}{dx} = (n-1) \sinh^{n-2} x + n \sinh^n x\). [3]
The integral \(I_n\) is defined by \(I_n = \int_0^{\operatorname{arsinh} 1} \sinh^n x \, dx\), \(n \geq 0\).
  1. Using the result in part (a), or otherwise, show that $$nI_n = \sqrt{2} - (n-1)I_{n-2}, \quad n \geq 2$$ [2]
  2. Hence find the value of \(I_4\). [4]