Integrating odd powers of secant (such as sec^3) combines the Pythagorean trigonometric identity (sec^2 = 1 + tan^2) with integration by parts, producing, after the by-parts step, an expression that reproduces the original integral with an opposite sign; algebraically solving this self-referential equation (adding the reproduced integral to both sides and dividing by two) yields the closed-form antiderivative. This belongs to integral calculus, within the techniques for integrating trigonometric functions, and includes the complementary strategy of applying trigonometric identities (such as the double-angle identity sin(2u) = 2 sin(u) cos(u)) to simplify an integrand into a directly recognizable standard form before integrating.
Integrating odd powers of secant (such as sec^3) combines the Pythagorean trigonometric identity (sec^2 = 1 + tan^2) with integration by parts, producing, after the by-parts step, an expression that …