Filon's Integration Formula

$\int_{x_0}^{x_n} f(x)\cos(tx)\,dx =h\{\alpha(th)[f_{2n}\sin(tx_{2n})-f_0\sin(tx_0)]$
$+\beta(th)C_{2n}+\gamma(th)C_{2n-1}+{\textstyle{2\over 45}}th^4 S_{2n-1}'\}-R_n,\quad$	(1)

where

$\displaystyle C_{2n}$	$\textstyle =$	$\displaystyle \sum_{i=0}^n f_{2i}\cos(tx_{2i})-{\textstyle{1\over 2}}[f_{2n}\cos(tx_{2n})+f_0\cos(tx_0)]$
			(2)
$\displaystyle C_{2n-1}$	$\textstyle =$	$\displaystyle \sum_{i=1}^n f_{2i-1}\cos(tx_{2i-1})$	(3)
$\displaystyle S_{2n-1}'$	$\textstyle =$	$\displaystyle \sum_{i=1}^n f^{(3)}_{2i-1} \sin(tx_{2i-1})$	(4)
$\displaystyle \alpha(\theta)$	$\textstyle =$	$\displaystyle {1\over\theta}+{\sin(2\theta)\over 2\theta^2}-{2\sin^2\theta\over\theta^3}$	(5)
$\displaystyle \beta(\theta)$	$\textstyle =$	$\displaystyle 2\left[{{1+\cos^2\theta\over\theta^2}-{\sin(2\theta)\over\theta^3}}\right]$	(6)
$\displaystyle \gamma(\theta)$	$\textstyle =$	$\displaystyle 4\left({{\sin\theta\over\theta^3}-{\cos\theta\over\theta^2}}\right),$	(7)

Tukey, J. W. In On Numerical Approximation: Proceedings of a Symposium Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, April 21-23, 1958 (Ed. R. E. Langer). Madison, WI: University of Wisconsin Press, p. 400, 1959.