| One of the major topics in the study of Integral Calculus is
evaluating integrals of powers of trigonometric functions.
Conventional methods usually use trigonometric identities
to transform integrands into forms where integration
formulas can be directly applied. These methods are
usually long, tedious, and time consuming. Since many
students are not patient enough to deal with lengthy and
complicated solutions, the objective of this paper was to
derive direct integration formulas in evaluating integrals of
powers of cotangent, and to develop simplified “one-liner”
solution for the integral.
Expository research method was used by first deriving a
reduction formula for cotangent. The reduction formula
was repeatedly applied to the integral of the nth power of
cotangent until the following formulas, for odd and even
powers, respectively, were derived:
Since the derivation process involved recursive relations,
the coefficients and exponents of the derived formulas
showed certain patterns and sequences which were used
as the basis for developing simpler algorithms. Comparison between the conventional method and the new
method revealed that the new method was simpler and
easier to use. The tiresome repetitions of applying the
reduction formula, or expansions of identities using the
conventional methods were eliminated. Integrals can be
evaluated directly since the procedure simply involved
coefficients and exponents of the integrand. It is suggested
that the same concept be extended in evaluating integrals
of powers of other trigonometric functions. |
| J. Stewart, Calculus: Concepts and Contexts, 4ed, Cengage,2012,
ch 5, pp 389-390
J. Hass, M. Weir, and G. Thomas, University Calculus, Pearson,
2014, ch8, pp. 439-440.
D. Varberg, E. Purcell and S. Rigdon, Calculus Early
Transcendentals, 1st ed., Pearson, 2014, ch 7, pp. 385-386.
K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for
Physics and Engineering, Cambridge University Press, 2010,
J. Stewart, Calculus: Early Transcendentals, 7ed, Cengage, 2014,
ch.7, pp467-468. |