(a)

The integral is .

The above integral is proper because the function  is a smooth continuous function in the entire integration region .

(b)

The integral is .

The above integral is improper because the function  is not continuous at point .

Therefore, convergence should be analyzed in the sense of a limit of proper integrals with finite integration ranges

i.e, .

(c)

The integral is .

Rewrite the integral as .

The above integral is improper because the integrand  is not continuous at point  and .

Therefore, the integral is improper and should be understood in the sense of a limit of proper integrals with finite integration ranges, i.e, .

(d)

The integral is .

.

The above integral is improper because upper integration limit is infinite.

This integral should be defined as a limit of proper integrals with finite integration range.

(a) Proper.

(b) Improper.

(c) Improper.

(d) Improper.