Suitable Gauss and Filon type quadrature methods for some highly oscillatory integrals

By the traditional quadrature rules/methods such as Newton-Cotes, Gauss-Legendre, or Clenshaw-Curtis, with the use of an acceptable number of quadrature nodes, it is not possible to obtain accurate approximations to integrals of the form(A)$ \int_0^1{f(x) sin{(\omega g(x))}dx}$ or $\int_0^1{f(x) cos{(\omega g(x))}dx}$in general, where $\omega$ and $r$ are given positive numbers, $g(x)=1/{x^r}$ and $f$ is a sufficiently smooth function on the interval $[0,1]$.In this work, the efficient quadrature methods (Filon, Levin, Asymptotic ) to compute highly oscillatory integrals are investigated and the Gauss quadrature rules developed at previous studies, which give very accurate results for (A), are generalized to compute the integrals (A) with $g(x)=\frac{1}{(x-x_1)^{r1}(x-x_2)^{r2}}$ accurately. In addition, appropriate Filon-type methods are presented for the computation of integrals obtained by the change of variable $x=1/t$ in (A).Key words: Gauss quadrature methods, Highly oscillatory integrals, Filon-type methods, Levin-type methods, Asymptotic methods.