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