Hyperbolic functions/integration

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In summary, the conversation discusses how to solve the integral from 5 to 3 of the square root of x^2 - 9 with respect to x. The conversation mentions using hyperbolic functions and a substitution of x = 3coth(theta) to solve the integral. The final solution is found using the substitution x = 3 cosh y.
  • #1
Zoe-b
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Homework Statement


Show that: (I don't have any way of showing it properly so have written it in words)

'The integral from 5 to 3 of (sq root (x^2 - 9) ) with respect to x' = 10 - (9/2) * ln 3

I hope that makes sense!

Homework Equations


Well I'm not sure.. I *think* that the fact the integral of 1/(sq root (x^2 - a^2)) is arcosh (x/a) is relevant.


The Attempt at a Solution


This is part of the 'hyperbolic functions' section on my exam; so the first thing I know is that they will come into it somehow. I cannot solve this by inspection, or by simply comparing it to all the formulae on my sheet.. I also tried to solve by parts (using difference of two squares to give me a root multiplied by another root) but found that this still gave me something I do not know how to integrate. I know if it was 1/(sq root (x^2 - 9)) it would be easy; but cannot work out how the integral of a particular thing and the integral of 1/it are related.

Thanks in advance.
 
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  • #2
Zoe-b said:

Homework Statement


Show that: (I don't have any way of showing it properly so have written it in words)

'The integral from 5 to 3 of (sq root (x^2 - 9) ) with respect to x' = 10 - (9/2) * ln 3

I hope that makes sense!

Homework Equations


Well I'm not sure.. I *think* that the fact the integral of 1/(sq root (x^2 - a^2)) is arcosh (x/a) is relevant.


The Attempt at a Solution


This is part of the 'hyperbolic functions' section on my exam; so the first thing I know is that they will come into it somehow. I cannot solve this by inspection, or by simply comparing it to all the formulae on my sheet.. I also tried to solve by parts (using difference of two squares to give me a root multiplied by another root) but found that this still gave me something I do not know how to integrate. I know if it was 1/(sq root (x^2 - 9)) it would be easy; but cannot work out how the integral of a particular thing and the integral of 1/it are related.

Thanks in advance.
It is true that [itex]cosh^2(\theta)- sinh^2(\theta)= 1[/itex] so, dividing both sides by [itex]cosh^(\theta)[/itex], [itex]1- tanh^2(\theta)= csch^2(\theta)[/itex] . That is, if you let [itex]x= 3coth(\theta), [itex]9- x^2= 9- 9coth^2(\theta)= 9csch^2(\theta)[/itex] so [itex]\sqrt{9- x^2}= \sqrt{9csch^2(\theta)}= 3csch(\theta)[/itex]. Of course, [itex]3(coth(\theta))'= 3(1- coth(\theta))[/itex] so [itex]dx= 3(1- coth(\theta))d\theta[/itex].
The integral becomes
[tex]9\int coth(\theta)(1- coth(\theta)d\theta[/tex]
 
  • #3
Don't think arccos is relevant. . .

Hmm I don't think a hyperbolic substition is necessary here . ..need to check my calculations ... .:eek:

//Argh forgot to square the minus// Nevermind::redface:
 
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  • #4
Right. eek.. I really didn't follow that! sorry. right..

what does [itex] mean?? and how did you get from sq root(x^2 - 9) to sq root (9 - x^2)? without getting into complex numbers..
 
  • #5
Zoe-b said:
Right. eek.. I really didn't follow that! sorry. right..

what does i tek mean?? and how did you get from sq root(x^2 - 9) to sq root (9 - x^2)? without getting into complex numbers..



I don't know how he did that, because that's what's worrying me each time I do this integral I get a complex function ... *not having a good day:(*

[ ite k or latek" is just a thing for typing lovely equations, e.g [tex]\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
 
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  • #6
Well I tried using the substitution x = 3coth theta... though I'll call theta y to make it easier to type.
I then get x^2 - 9 = 9coth^2 y - 9
root (x^2 - 9) = 3(cosech y)

therefore the original question is:
the integral from x=5 to x=3 of 1/(3 (cosech y)) dx.

dx/dy = -3(cosech y) ... I think, not so sure on my maths there. now I get stuck again as this seems to cancel to give no variable to integrate..
 
  • #7
Zoe-b said:
Well I tried using the substitution x = 3coth theta... though I'll call theta y to make it easier to type.
I then get x^2 - 9 = 9coth^2 y - 9
root (x^2 - 9) = 3(cosech y)

therefore the original question is:
the integral from x=5 to x=3 of 1/(3 (cosech y)) dx.

dx/dy = -3(cosech y) ... I think, not so sure on my maths there. now I get stuck again as this seems to cancel to give no variable to integrate..

...oopps... never mind that

[tex] \frac{dx}{d\theta}=3 cosch^2 ({\theta}) [/tex] .. .



Starting over grrr


Yay I've finally got it! ... I think
 
Last edited:
  • #8
I found a solution online (the cheats way out I know!).. but I'll do it properly when I get time. Its easier to use a different substitution (x = 3 cosh y, if I remember rightly).
Thanks anyway.
 
  • #9
sorry HallsofIvy : 1- tanh^2 (theta)= sech^2(theta) !
thanks
 

1. What are hyperbolic functions and how are they different from trigonometric functions?

Hyperbolic functions are mathematical functions that are related to the hyperbola. They are commonly denoted as "sinh", "cosh", "tanh", etc. and are defined in terms of exponential functions. These functions have similar properties to trigonometric functions, but they differ in their shape and behavior.

2. How are hyperbolic functions used in real-world applications?

Hyperbolic functions are used in a variety of fields, including physics, engineering, and mathematics. They are commonly used to model the shape of hanging cables, calculate the trajectory of a projectile, and solve differential equations.

3. What is the relationship between hyperbolic functions and integration?

Hyperbolic functions are closely related to integration, as they can be expressed in terms of integrals. In fact, the inverse hyperbolic functions, such as "arcsinh" and "arctanh", are defined as integrals of certain hyperbolic functions. This relationship is useful in solving integrals involving hyperbolic functions.

4. Can hyperbolic functions be graphed?

Yes, hyperbolic functions can be graphed using a graphing calculator or software. They have a similar shape to trigonometric functions, but they are more spread out and have a steeper curve. The graphs of hyperbolic functions can also be transformed using different parameters, similar to trigonometric functions.

5. Are there any special properties of hyperbolic functions?

Yes, hyperbolic functions have several special properties that make them useful in mathematics. For example, they satisfy certain identities, such as the hyperbolic Pythagorean identity and the hyperbolic addition formulas. They also have a close relationship with complex numbers, which allows for the use of hyperbolic functions in complex analysis.

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