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

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.