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ISSN : 1226-0657

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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: The Pure and Applied Mathematics / 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
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 $\boldsymbol\int_0^\infty[\sin^{2m+1}(\alpha x)]/(x^{2n+1})dx$. Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics, 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

Journal of the Korean Society of Mathematical Education Series B: The Pure and Applied Mathematics

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