It is the very symbol of "indefinite integral" that is flawed and confusing. It should be removed and kept only as a "guilt practice", like treating $dy/dx$ as a real fraction and things like that.
I came across this statement as a well-received comment on another question. I'm interested in understanding what the reasons are for this position, primarily because I'm curious about why something I've been taught since childhood might be wrong.
Also, as I'm starting to study mathematics at postgraduate level I'm seeing definite integration used more often. For example initial conditions are often applied directly as in $y(0)=0, f(y)\,dy/dx = g(x) \implies \int_0^yf(u)\,du = \int_0^x g(v)dv$, and I'm having to get used to seeing solutions written out with embedded definite integrals where analytic results can't be found. With this comment in mind I'm wondering if a more parsimonious approach would be better, dropping indefinite integrals altogether.