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?