Question

what is the main difference between proper integrals & improper integrals?

Answer 1

Basically, as far as I gather, improper integrals are those which involve infinities in one of their endpoints (a,b etc.) are +/- infinity (this includes asymptotes (infinite on the y-axis) and when a/b= infinity (infinite on the x-axis)).
That's as far as my meagre knowledge stretches, hope it helps.

Answer 2

An integral becomes improper for two reasons:

i) Either the upper or lower limit is infinite
ii) If a point of discontinuity exists on the interval is being integrated.

For example, the following is a improper integral because it's upper bound is infinite:

∫ e^(-x) dx (from x=0 to infinity).

This next one is improper due to the discontinuity at x = 0:

∫ 1/√x dx (from x=0 to 1).

Of course, there are integrals that are improper for both reasons (having a discontinuity on the integration interval AND having an un-bounded end-point).

Answer 3

so proper inegrals are those without intervals?