¿Quién se dio cuenta de $\int \frac 1x dx =\ln(x)+c$?

Question

¿Quién descubrió el $\int \frac 1x dx=\ln(x)+c$ no obvio? ¿Hubo series de poder involucradas? La serie se ve similar en lados opuestos de 1:

$$ \frac 1x =\sum_{n=0}^\infty (-1+x)^n \text{ for } |x-1|

$$ \ln(x) = \ln(-1+x)-\sum_{k=1}^\infty \frac{(-1)^k}{k(-1+x)^k} \text{ for } |x-1|>1 $$

$$ \ln(x) = -\sum_{k=1}^{\infty}\frac{(-1)^k(-1+x)^k}k \text{ for } |x-1|

Answer 1

Es una combinación de ideas. La primera es la propiedad de que la cuadratura de la hipérbola satisface $f(xy)=f(x)+f(y)$. Esto parece haber sido descubierto por Gregoire de Saint-Vincent en su Opus geometricum quadrature circuli et sectionum coni (1647).

En ella

Esto no es otra cosa que la ecuación

$$\int{1}^{xy}\frac{\text{d}t}{t}=\int{1}^{x}\frac{\text{d}t}{t}+\int_{1}^{y}\frac{\text{d}t}{t}$$ "Arithmetic series" means the addition on the right hand side, "geometric series" means the multiplication $xy$ on the left. Then there is the idea of logarithms to ease computations and the idea of using the most natural basis. The natural logarithms were first called hyperbolic logarithms. Logarithms used to be tabulated in the times of by-hand computations. Then, the name natural logarithm seems to come from Mercator in his Logarithmotechnia (1668). Apparently it was Euler the one who put together $\ln$ and $e$ in $\ln=\loge$; a constant hinted in a publication of Napier, and computed first by Jakob Bernoulli as $\lim{n\rightarrow \infty}(1+\frac{1}{n})^n$.