Exact value of integrals involving product of sine or cosine function

Abstract

By considering the number of all choices of signs $+$ and $-$ such that $\pm \alpha_1 \pm \alpha_2 \pm \alpha_3 \cdots \pm \alpha_n = 0$ and the number of sign $-$ appeared therein, this paper can give the exact value of $\int_{0}^{2\pi} \prod_{k=1}^{n} \sin (\alpha_k x) dx$. In addition, without using the Fourier transformation technique, we can also find the exact value of $\int_{0}^{\infty}\frac{(\cos\alpha x - \cos\beta x)^p}{x^q} dx$. These two integrals are motivated by the work of Andrican and Bragdasar in 2021, Andria and Tomescu in 2002, and Borwein and Borwein in 2001, respectively.