A closed form solution for an unorthodox trigonometric integral. (English)

Int. J. Math. Educ. Sci. Technol. 40, No. 6, 814-817 (2009).

Summary: A closed form solution for the trigonometric integral $\int\text{sec}^{2k+1}xdx$, $k=0,1,2,\dots$, is presented in this article. The result will fill the gap in another trigonometric integral $\int\text{sec}^{2m+1}x\tan^{2n}xdx$, which is neglected by most of the calculus textbooks due to its foreseeable unorthodox solution procedure comparing to the other three cases in $\int\text{sec}^mx$ when $m$ is even or $n$ is odd.