Can someone me with a quick integral?

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Discussion Overview

The discussion revolves around the integral of the function 1/((sinh[x])^2) with respect to x. Participants explore various methods for solving this integral, including substitutions and comparisons to trigonometric integrals. The scope includes mathematical reasoning and techniques related to integration.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests that rewriting the hyperbolic sine in exponential form could simplify the integral, potentially using substitution or partial fractions.
  • Another participant draws parallels between hyperbolic and trigonometric integrals, proposing that the antiderivative of 1/sinh might involve cosh/sinh.
  • A different approach mentioned involves breaking the integral into two fractions and integrating each separately.
  • One participant expresses uncertainty about their approach, indicating they reached a point involving the integral of sech(2x) but questions its correctness.

Areas of Agreement / Disagreement

Participants present multiple competing views on how to approach the integral, and there is no consensus on the best method or whether any proposed method is definitively correct.

Contextual Notes

Some participants' suggestions depend on the familiarity with hyperbolic functions and their relationships to trigonometric functions, which may not be universally understood. There are also unresolved steps in the proposed methods.

MaximumTaco
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I don't know how to do fancy symbols, but

Integral of 1/((sinh[x])^2) .dx

I got as far as 1/2 Int(sech(2x)). d(2x), is this the right approach?

Thanks .
 
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Your approach might work, but it seems to me like the integral could be easily handled by rewriting the hyperbolic sine in exponential form and then doing a substitution (or partial fractions (but that seems rather messy)).
 
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this is probably easy if you remember the basic trig integrals and their analogies with hyperbolic trig integrals. i.e. remember the derivative of tan is sec^2, and sec = 1/cos.

also the derivative of cot is -csc^2 and csc = 1/sin.

Hence by analogy, we should try cosh/sinh as the antiderivative of 1/sinh. does it work?
 
It write it in it’s exponent form then break it into two fractions and take each integral separately.
 

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