Question

integrate 1/x dx w.r.t x
answer should be log(x) or ln(x)
and why

Answer 1

because d/dx log(x) = 1/x,

∫dx/x = log(x)+C

is that why enough, or is there yet something missing?

And, maybe you haven't seen the old chestnut

∫d(cabin)/cabin = log cabin + C = houseboat