The significance of proving this integral is that it provides a valuable mathematical result that can be used in various applications. It also allows for a better understanding of the properties and behavior of the hyperbolic tangent function.
The upper limit of integration is infinity because the integrand, ${\tanh^2(t)\tanh(2t)\over t^2}$, is defined for all real values of $t$ and does not have any singularities or discontinuities at infinity. This allows for the integral to be evaluated over an infinite range, resulting in a more precise and accurate value.
The value of $2\ln 2$ is derived using various techniques from calculus, such as integration by parts and substitution. The specific steps and calculations involved in the derivation may vary, but the resulting value is found to be $2\ln 2$.
Yes, this integral can also be solved using other methods such as contour integration or the residue theorem. However, the result will still be the same, $2\ln 2$, as long as the calculations are done correctly.
This integral can be used in various fields such as physics, engineering, and economics. For example, it can be used to calculate the work done by an object moving with a velocity proportional to the hyperbolic tangent function, or to model population growth in economics using the hyperbolic tangent function.