There are (at least) three types of ``cosine integrals,'' denoted
,
, and
:
(1)  
(2)  
(3)  
(4)  
(5)  
(6) 
To compute the integral of an Even power times a cosine,
(7) 
(8) 
(9) 
(10) 
(11) 
(12) 

(13) 
(14) 
To find a closed form for an integral power of a cosine function,
(15) 
(16) 
(17) 
(18) 
(19) 
(20) 
Now, if is Even so , then
(21) 
(22) 
Now if is Odd so , then
(23) 
Now let ,
(24) 
(25) 
The infinite integral of a cosine times a Gaussian can also be done in closed form,
(26) 
See also Chi, Damped Exponential Cosine Integral, Nielsen's Spiral, Shi, Sici Spiral, Sine Integral
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Sine and Cosine Integrals.'' §5.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 231233, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342343, 1985.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Fresnel Integrals, Cosine and Sine Integrals.'' §6.79 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 248252, 1992.
Spanier, J. and Oldham, K. B. ``The Cosine and Sine Integrals.'' Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361372, 1987.
© 19969 Eric W. Weisstein