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I find the notation $\int f(x)dx$ for the indefinite integral of $f(x)$ on some interval $I$ is both suggestive and confusing. On the one hand, this notation is very suggestive when we calculate the indefinite integral either by the change of variable formula or the integration by parts formula. On the other hand, this notation looks very likely with the definite integral notation $\int_a^b f(x)dx$. But these two terms arise from different backgrounds, one to find the primitive while the other to find the area.

I want to know who introduced the term indefinite integral and the notation $\int f(x)dx$ and why?

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    $\begingroup$ Good question; you may want to check out History of Science and Mathematics in case it's already answered there. $\endgroup$ – user147263 Dec 15 '14 at 3:24
  • $\begingroup$ I believe Leibniz. Recall the definition of the definite integral, taking the limit of Riemann sums. Change $\lim \sum$ to $\int$, $f(x_i ^*)$ to $f(x)$, and $\Delta x$ to $dx$, and there you go! (Note that $\int$ is like an elongated S, for "sum"... $\endgroup$ – Forever Mozart Dec 15 '14 at 3:28
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    $\begingroup$ This is a good question, but for History of Math and Science Stack Exchange. $\endgroup$ – Mark Fantini Dec 15 '14 at 3:32
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Notation was introduced by Leibniz..Earlier he used $\overline{omn} l$ where omn stands for sum and $l$ for differences.Later he used $\int$ for $\overline{omn}$. It was elongated S from sum.

Before Newton and Leibniz mathematicians like Roberval has arrived at relation between area under the curve and anti-derivative[Fundamental theorem]. But they cannot state it explicitly. Both Newton and Leibniz discussed integral as anti-derivative and stated Fundamental theorem

It was Cauchy who developed the definition Definite integral as limit of sum in proper way.Riemann generalized Cauchy's method.For reference you can go through The Historical Development of Calculus by C H Edwards.

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  • $\begingroup$ Thanks so much! I find some useful information on this topics on the website: jeff560.tripod.com/i.html: The term INDEFINITE INTEGRAL is defined by Sylvestre-François Lacroix (1765-1843) in Traité du calcul différentiel et integral (Cajori a history of mathematics, 1919, page 272). Indefinite integral appears in English in 1828 in An Elementary Treatise on the Differential and Integral Calculus by Jean-Louis Boucharlat and Ralph Blakelock [Google print search]. $\endgroup$ – longtemps Dec 15 '14 at 4:16
  • $\begingroup$ @longtemps, i searched through A History of Mathematics by Cajory but did not find where he would attribute the term indefinite integral to Lacroix. I also looked into the electronic copy of the second volume of Traité du calcul différentiel et integral but did not notice this term to be introduced. As to An Elementary Treatise on the Differential and Integral Calculus, it is simply an English translation of the Traité. $\endgroup$ – Alexey yesterday

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