Solving Hyperbolic Integral 1/(1+cosh(x)) with Wolfram

In summary, the conversation is about solving the integral of 1/(1+cosh(x)). The speaker has used Wolfram to get the solution but is still having trouble understanding the second transformation used. They mention using tan x = t but not for hyperbolic functions. The person helping them then explains that the only substitution used is u=tanh(x/2) and suggests looking at the identities in hyperbolic trigonometry. They then proceed to show the working, using the double angle identity and Osbournes Rule. The speaker thanks them for their help.
  • #1
FelixHelix
28
0
Hi there. I've been trying to solve the integral of 1/(1+cosh(x)). I use Wolfram to give me a detailed solution but I still don't understand second transformations they use.

I've attached a a screen grab of the workings and hoped someone could run through it with me.

I've used the tan x = t but never for hyperbolic...

Thanks F
 

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  • #2
They use only one substitution: u=tanh(x/2). What's not clear about that ?
 
  • #3
I suppose I mean the second transformation. I can't see how you get there... What identities are at work...?
 
  • #5
Starting from the identity
[tex]\cosh x=\cosh^2 (x/2)+\sinh^2(x/2)=2 \cosh^2(x/2)-1,[/tex]
we get
[tex]\int \mathrm{d} x \frac{1}{1+\cosh x}=\frac{1}{2} \int \mathrm{d} x \frac{1}{\cosh^2(x/2)}=\tanh(x/2).[/tex]
Of course one should now that
[tex]\frac{\mathrm{d}}{\mathrm{d} x} \tanh x=\frac{1}{\cosh^2 x}.[/tex]
 
  • #6
Ahhh, I see now. Double angle identity and then using Osbournes Rule... easy when you see it!

Thanks vanhees71 for taking the time to show the working - It's much appreciated...

FH
 

1. How do I use Wolfram to solve hyperbolic integrals?

Wolfram is a powerful computational tool that can solve a wide range of mathematical problems, including hyperbolic integrals. To use Wolfram to solve a hyperbolic integral, simply input the integral into the Wolfram Alpha search bar and click "solve". Wolfram will then provide you with the solution and step-by-step instructions on how to solve it.

2. What is a hyperbolic integral?

A hyperbolic integral is a mathematical function that involves the hyperbolic cosine (cosh) or hyperbolic sine (sinh) of a variable. It is often used to solve problems in physics and engineering, particularly in the study of electromagnetic fields and vibrations.

3. Can Wolfram solve any type of hyperbolic integral?

Wolfram can solve most types of hyperbolic integrals, but there may be some rare or complex integrals that it cannot solve. In these cases, Wolfram will provide an approximation or numerical solution.

4. What is the syntax for inputting a hyperbolic integral into Wolfram?

To input a hyperbolic integral into Wolfram, use the following syntax: integral of 1 over open parentheses 1 plus cosh of x close parentheses, where x is the variable of integration. You can also use the shorthand notation "integrate 1/(1+cosh(x)) dx".

5. Are there any tips for using Wolfram to solve hyperbolic integrals?

When using Wolfram to solve hyperbolic integrals, it is important to double-check the solution and make sure it is in the correct form. Wolfram may use different notations or rearrange terms in the solution, but the end result should be equivalent to the original integral. It is also helpful to familiarize yourself with the various options and settings in Wolfram to customize the solution to your needs.

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