ON THE INTEGRATION AND THE REDUCTION OF BINOMIAL DIFFERENTIALS AND OF ANY OTHER DIFFERENTIAL FORMULAS OF THE SAME TYPE

Dennis M. Cates ()

Chapter Chapter 29 in Cauchy's Calcul Infinitésimal, 2019, pp 155-160 from Springer

Abstract: Abstract Let $$ a, b, a_1, b_1, \lambda , \mu , \nu $$ be real constants, y a variable quantity, and let us make $$y^{\lambda }=x. \ $$ The expression $$(ay^{\lambda }+b)^{\mu } dy, $$ in which dx has for a coefficient a power of the binomial $$ay^{\lambda }+b, $$ will be what we call a binomial differential, and the indefinite integral $$\begin{aligned} \int {(ay^{\lambda }+b)^{\mu } dy}=\frac{1}{\lambda }\int {(ax+b)^{\mu }x^{\frac{1}{\lambda }-1} dx}\end{aligned}$$ will be the product of $$\frac{1}{\lambda }$$ with another integral included in the general formula $$\begin{aligned}\int {(ax+b)^{\mu }(a_1x+b_1)^{\nu } dx}, \end{aligned}$$ which we will now occupy ourselves.

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_29

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