There are two types of “sine integrals” commonly defined,

(1)

(2)

(3)

(4)

(5)

(6)

To compute the integral of a sine function times a power

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

General integrals of the form

(15)

See also Chi, Cosine Integral, Exponential Integral, Nielsen’s Spiral, Shi, Sici Spiral, Sinc Function

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. 231-233, 1972.
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342-343, 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. 248-252, 1992.
Spanier, J. and Oldham, K. B. “The Cosine and Sine Integrals.”
Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361-372, 1987.

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 342-343, 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. 248-252, 1992.

Spanier, J. and Oldham, K. B. “The Cosine and Sine Integrals.” Ch. 38 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 361-372, 1987.