Interesting definite and improper integralsINTRODUCTIONMost of the integrals that are analytical have proved to have interesting applications in the outsideworld. Yet, there are integrals which have, important applications in the real world, but are counter-intuitive.These integrals often do not have an analytical solution in terms of standard functions, buthave a finite value for the improper integral. For instance the integrals,∫0∞(lnx)2(x2+1)2dxand∫0∞lnx(x2+1)dx, are not analytical in terms of standard functions, yet, astonishingly, a finite value isobtained for the improper integrals ranging from 0 to infinity. In contrast, an analytical solution to theintegral∫tannx dx, where n is a real number, could be found by employing special methods.However, it is not at all obvious how such an integral could produce an analytical solution. Theseobservations left me baffled and it came to thought that such integrals would prove interesting toinvestigate.In this exploration I would, in the first section, discuss the integral of∫tannx dx, specifically whenn is a rational number, and attempt to find a value for the improper integral from 0 to infinity. In thesecond section I would prove that an analytical solution forex2andcosxxcannot be arrived at,in terms of standard functions, but I will attempt to provide an alternate solution using series
expansions. Consequently, I would explore different methods to find the value for the improperintegrals.SECTION1Problem 1let’s consider the functionf(x)=tannx, where n∈ℝ.For n=1, the standard tangent function could be integrated to give–ln|cosx|.This could besimplified to giveln|secx|. However, when n approaches rational numbers; the integral turnsrather complex.Let us then let n be12. The integral could be expressed as∫√tanx dx.The substitution for suchan integral is very hard to deduce. In order to evaluate this integral we will define two supplementaryfunctions,I1andI2, such that the sum or the difference would result in the integral we areinterested in.