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