Definition of an improper integral :

(i)

If  is continuous on  and is discontinuous at , then  if this limit exists (as a finite number).

(ii)

If  is continuous on  and is discontinuous at , then  if this limit exists (as a finite number).

(a)

The integral is .

The function is .

The given integral is improper because  has the vertical asymptote at .

Thus, the infinite discontinuity occurs at the left end point of .

(b)

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.

(c)

The integral is .

, where  is arbitary.

The above integral is improper because bith upper and lower integration limits are infinite.

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

(d)

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.

i.e, .

(a) Infinite discontinuity.

(b) Infinite interval.

(c) Infinite interval.

(d) Infinite discontinuity.