INTEGATING POLYNOMIALS IN SECANT AND TANGENT
JONATHAN P. MCCAMMOND
This note provides a relatively painless way to integrate arbitrary polynomials in secant and tangent without ever invoking integration by parts or anything beyond elementary polynomial and trigonometric identities. The techniques involved also introduce students to some of the ideas behind the construction of Laurent polynomials, although the manner in which they do so is rather indirect. We begin with a theorem which covers almost all of the possibilities.
Theorem 1.For every polynomial P(s, t) in two variables, there are polynomials F and G in one variable and a constant C such that
∫▒〖p(sec〖x ,tan〖x 〗 〗)〗 sec〖x dx=F(u)- G(v)+ C ln(u)+ C 〗
where u = sec x + tan x and v = sec x – tan x.
Proof. Once we