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

  • Dennis M. CatesEmail author
Chapter

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.

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sun CityUSA

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