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?
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.