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

Qi, Feng; Qi, Feng; Luo, Qiu-Ming; Luo, Qiu-Ming; Guo, Bai-Ni; Guo, Bai-Ni

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P-ISSN3059-0604
E-ISSN3059-1309
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Vol.11 No.3

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

Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics / Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics, (P)3059-0604; (E)3059-1309

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

Qi, Feng
Luo, Qiu-Ming
Guo, Bai-Ni

Qi,, F. , Luo,, Q. , & Guo,, B. (2004). Evaluations of the Improper Integrals $\boldsymbol\int_0^\infty[\sin^{2m}(\alpha x)]/(x^{2n})dx$ and $\boldsymbol\int_0^\infty[\sin^{2m+1}(\alpha x)]/(x^{2n+1})dx$, 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 ${\int}_0^{\infty}$[sin$^{2m}({\alpha}x)]/(x^{2n})dx$ AND ${\int}_0^{\infty}$[sin$^{2m+1}({\alpha}x)]/(x^{2n+1})dx$ are established, where m $\geq$ n are all positive integers and $\alpha$$\neq$ 0.

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

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Article Contents

Vol.11 No.3

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

Journal of the Korean Society of Mathematical Education Series B: Theoretical Mathematics and Pedagogical Mathematics