You multiply by dt (or divide by the sample time Ts) to take into account that the algorithm may run at different speeds in for example a micro controller. You are in essence approximating the integral of the error function e(t), i.e., finding the area under its graph by taking the sum of a series of columns with hight e(k) (err) and width dt.

To illustrate why dt is needed.

Say you are running your PID algorithm every 100 ms, in one second you will sum the error 10 times.

Now say you are running at ten times that speed, sampling and running the algorithm every 10 ms. Now you will sum the error 100 times. If you don't take dt into consideration the integral part will now be 10 times larger than it was when the algorithm was running at 100 ms.

Also, you would typically multiply the error by a gain factor, called the integral gain K_I. This will take care of your worries of a too low I-value.

Hope this helps to clarify!

As already stated, the integrator gain K_i is missing in the formulas.

As for the "missing" dt: the second formula works very well if dt is constant, the first works when dt varies significantly. Though a varying dt is capable to raise some "issues" beyond that. If dt variation is only small (eg. some "jitter"), the second formula is the better one - concatenating some "generic K_i" and the dt influence into a single K_i constant.