Find formula of a function

[Thanasis]

I’m hard stuck with the following problem. Do you have any idea?
Let F: R ->R be a continuous function for which
e^F(x)-e^-F(x)=2x for every x in R
Find the formula of F(x).

 

[MarkFL]

I would begin by writing the relation in the following form:

[MATH]x=\sinh(F(x))[/MATH]
Can you proceed?

 

[Thanasis]

Nevermind. I just found what I should do. Here is the answer for anyone being curious about it.

 

[MarkFL]

Nevermind. I just found what I should do. Here is the answer for anyone being curious about it.

I don't see an answer in your post, but I would continue, and write:

[MATH]F(x)=\arsinh(x)[/MATH]

 

[Thanasis]

Here it is

 

[MarkFL]

Yes, it can be shown that:

[MATH]\arsinh(x)=\ln\left(x+\sqrt{x^2+1}\right)[/MATH]
Note: there's no need for absolute value brackets around the argument of the natural log function, as it will be positive for all real \(x\).

 

[Thanasis]

Yeah I noticed that it is positive.