Proving $\int_{0}^{\infty}{\tanh^2(t)\tanh(2t)\over t^2}\mathrm dt=2\ln 2$

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SUMMARY

The integral $\int_{0}^{\infty}{\tanh^2(t)\tanh(2t)\over t^2}\mathrm dt$ evaluates to $2\ln 2$. This conclusion is derived from advanced calculus techniques involving hyperbolic functions. The discussion emphasizes the importance of understanding the properties of $\tanh(t)$ and its behavior at infinity. The integral's convergence and the application of specific mathematical identities are crucial for the proof.

PREREQUISITES
  • Understanding of hyperbolic functions, specifically $\tanh(t)$
  • Knowledge of improper integrals and convergence criteria
  • Familiarity with calculus techniques for evaluating integrals
  • Experience with mathematical identities and transformations
NEXT STEPS
  • Study the properties of hyperbolic functions, focusing on $\tanh(t)$ and $\tanh(2t)$
  • Learn techniques for evaluating improper integrals, particularly those involving limits
  • Explore mathematical identities related to hyperbolic functions
  • Investigate the convergence of integrals involving trigonometric and hyperbolic functions
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced integral calculus and hyperbolic function properties.

Tony1
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Prove that,

$$\int_{0}^{\infty}{\tanh^2(t)\tanh(2t)\over t^2}\mathrm dt=\color{blue}{2\ln 2}$$
 
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Hi Tony. I recommend you read up on our guidelines for posting challenge problems, found https://mathhelpboards.com/challenge-questions-puzzles-28/guidelines-posting-answering-challenging-problem-puzzle-3875.html.
 

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