Main difference:
A proper integral has finite limits and the function is finite on the interval.
An improper integral has an infinite limit or the function becomes infinite somewhere.
Why limits are needed:
Because we cannot directly integrate to infinity or at a point where the function blows up, so we use limits to see if the value approaches a finite number.
Physical example:
In electrostatics, the electric field of a point charge involves terms
Integrating this near or to infinity gives improper integrals.
all improper integrals with infinity diverge
A proper integral has finite limits and the function is finite on the interval.
An improper integral has an infinite limit or the function becomes infinite somewhere.
Why limits are needed:
Because we cannot directly integrate to infinity or at a point where the function blows up, so we use limits to see if the value approaches a finite number.
Physical example:
In electrostatics, the electric field of a point charge involves terms
Integrating this near or to infinity gives improper integrals.
all improper integrals with infinity diverge