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November 2021 Complete Formulas for $\mathbf{\int x^{\pm n} (a^2 - x^2)^{\pm \frac{1}{2}} \ dx}$
Scott H. Demsky
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Missouri J. Math. Sci. 33(2): 151-157 (November 2021). DOI: 10.35834/2021/3302151

Abstract

We present complete antiderivative formulas not reduction formulas) for $ \int x^{n} \sqrt{a^2 - x^2} \ dx $, $ \int x^{n}/\sqrt{a^2 - x^2} \ dx $, $ \int \sqrt{a^2 - x^2}/x^{n} \ dx $, and $ \int 1/(x^{n}\sqrt{a^2 - x^2}) \ dx $, where $ n $ is any nonnegative integer. We shall provide a complete proof of one formula, the proofs of the others being quite similar.

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Scott H. Demsky. "Complete Formulas for $\mathbf{\int x^{\pm n} (a^2 - x^2)^{\pm \frac{1}{2}} \ dx}$." Missouri J. Math. Sci. 33 (2) 151 - 157, November 2021. https://doi.org/10.35834/2021/3302151

Information

Published: November 2021
First available in Project Euclid: 30 November 2021

Digital Object Identifier: 10.35834/2021/3302151

Subjects:
Primary: 26A36

Keywords: antiderivative , calculus , Indefinite integral , Integration , radical , reduction formula , square root , table of integrals , trigonometric substitution

Rights: Copyright © 2021 University of Central Missouri, School of Computer Science and Mathematics

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Vol.33 • No. 2 • November 2021
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