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
will be the product of \(\frac{1}{\lambda }\) with another integral included in the general formula
which we will now occupy ourselves.
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Notes
- 1.
In the 1823 edition the third expression reads, \( (ax+b)^{\pm \frac{m}{n}}(a_1x+b_1)^{\pm l \mp \frac{m}{n}} dx.\)
- 2.
The 1899 text uses the phrase une ou deux de suite meaning once or twice in sequence, instead of Cauchy’s original une ou plusieurs fois de suite translated here.
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Cates, D.M. (2019). ON THE INTEGRATION AND THE REDUCTION OF BINOMIAL DIFFERENTIALS AND OF ANY OTHER DIFFERENTIAL FORMULAS OF THE SAME TYPE.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_29
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DOI: https://doi.org/10.1007/978-3-030-11036-9_29
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