Evaluations of the Improper Integrals $\boldsymbol\int_0^\infty[\sin^{2m}(\alpha x)]/(x^{2n})dx$ and $\boldsymbol\int_0^\infty[\sin^{2m+1}(\alpha x)]/(x^{2n+1})dx$

한국수학교육학회지시리즈B:순수및응용수학 / Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, (P)1226-0657; (E)2287-6081

2004, v.11 no.3, pp.189-196

Qi, Feng (Department of Applied Mathematics and Informatics, Research Institute of Applied Mathematics, Henan Polytechnoc University)
Luo, Qiu-Ming (Department of Mathematcis, Jiaozuo University)
Guo, Bai-Ni (Department of Applied Mathematics, and Informatics, Research Institute of Applied Mathematics, Henan Polytechnic University)

Qi, Feng, Luo, Qiu-Ming, & Guo, Bai-Ni. (2004). EVALUATIONS OF THE IMPROPER INTEGRALS <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m}({\alpha}x)]/(x^{2n})dx$</TEX> AND <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$</TEX>. 한국수학교육학회지시리즈B:순수및응용수학, 11(3), 189-196.

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Abstract

In this article, using the L'Hospital rule, mathematical induction, the trigonometric power formulae and integration by parts, some integral formulae for the improper integrals <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m}({\alpha}x)]/(x^{2n})dx$</TEX> AND <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$</TEX> are established, where m <TEX>$\geq$</TEX> n are all positive integers and <TEX>$\alpha$</TEX><TEX>$\neq$</TEX> 0.

keywords: evaluation, improper integral, integral formula, inequality, integration by parts, L′Hospital rule, mathematical induction

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논문 상세

Vol.11 No.3

EVALUATIONS OF THE IMPROPER INTEGRALS <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m}({\alpha}x)]/(x^{2n})dx$</TEX> AND <TEX>${\int}_0^{\infty}$</TEX>[sin<TEX>$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$</TEX>

Evaluations of the Improper Integrals $\boldsymbol\int_0^\infty[\sin^{2m}(\alpha x)]/(x^{2n})dx$ and $\boldsymbol\int_0^\infty[\sin^{2m+1}(\alpha x)]/(x^{2n+1})dx$

Abstract

한국수학교육학회지시리즈B:순수및응용수학