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- TL;DR Summary
- Determine the area enclosed by the curve ##y=x \text{ csch}(x+y)## and the positive ##x##-axis

The title and summary pretty much say it all. I was wondering if it's possible to accurately determine the area enclosed by the curve ## y=x \text{ csch}(x+y)## and the ##x##-axis?

I first tried solving for ##y## and then ##x##, however it doesn't appear possible to solve for either variable. I then recognised that the equation can be represented as a nested function, such that: $$f(x)=x \text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+...))))$$

Through this I was able to use Desmos Graphing calculator to help me estimate the enclosed area to by integrating ##f(x)## from ##0## to ##\infty##. However, the imbeded nature of the function made it hard to properly find a definitive solution and I could only gather that the area appears to be approaching ##1.5## square units, although this might be inaccurate and it could be simply approaching a number near ##1.5##.

How would I go about accurately determining this area?

Any help is greatly appreciated.

I first tried solving for ##y## and then ##x##, however it doesn't appear possible to solve for either variable. I then recognised that the equation can be represented as a nested function, such that: $$f(x)=x \text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+x\text{ csch}(x+...))))$$

Through this I was able to use Desmos Graphing calculator to help me estimate the enclosed area to by integrating ##f(x)## from ##0## to ##\infty##. However, the imbeded nature of the function made it hard to properly find a definitive solution and I could only gather that the area appears to be approaching ##1.5## square units, although this might be inaccurate and it could be simply approaching a number near ##1.5##.

How would I go about accurately determining this area?

Any help is greatly appreciated.

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