# Exact value of integrals involving product of sine or cosine function

## DOI:

https://doi.org/10.53733/235## Keywords:

product of sine integral, product of difference of cosine integral, integer sequence## 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.

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## Published

12-10-2022

## How to Cite

*New Zealand Journal of Mathematics*,

*53*, 51–61. https://doi.org/10.53733/235

## Issue

## Section

Articles