Complete Formulas for $\mathbf{\int x^{\pm n} (a^2 - x^2)^{\pm \frac{1}{2}} \ dx}$

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.