Bode's Rule

$$\int_{x_1}^{x_5} f(x)\,dx = {\textstyle{2\over 45}} h (7f_1+... ...2+12f_3+32f_4+7f_5)-{\textstyle{8\over 945}} h^7 f^{(6)}(\xi).$$

See also Hardy's Rule, Newton-Cotes Formulas, Simpson's 3/8 Rule, Simpson's Rule, Trapezoidal Rule, Weddle's Rule

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 886, 1972.