Integral of radical trigonometric functions revisited

TL;DR

This paper reviews various methods for integrating radical trigonometric functions, highlighting their applications in calculating the length of cardioids in polar coordinates, and providing a comprehensive revisit of this classical integral.

Contribution

It introduces multiple integration techniques for radical trigonometric functions and discusses their applications in geometric calculations like cardioid lengths.

Findings

Multiple methods for integrating radical sine and cosine functions

Application of these integrals in calculating cardioid lengths

Enhanced understanding of classical integral techniques

Abstract

This article revisits an integral of radical trigonometric functions. It presents several methods of integration where the integrand takes the form $\sqrt{1 \pm \sin x}$ or $\sqrt{1 \pm \cos x}$. The integral has applications in Calculus where it appears as the length of cardioid represented in polar coordinates.