Page 319 - Schaum's Outline of Theory and Problems of Advanced Calculus

310 IMPROPER INTEGRALS [CHAP. 12

IMPROPER INTEGRALS OF THE SECOND KIND
If f ðxÞ becomes unbounded only at the end point x ¼ a of the interval a @ x @ b, then we define
b b
ð ð
f ðxÞ dx ¼ lim f ðxÞ dx ð4Þ
a !0þ aþ
and define it to be an improper integral of the second kind. If the limit on the right of (4) exists, we call
the integral on the left convergent; otherwise, it is divergent.
Similarly if f ðxÞ becomes unbounded only at the end point x ¼ b of the interval a @ x @ b, then we
extend the category of improper integrals of the second kind.

ð b ð b
f ðxÞ dx ¼ lim f ðxÞ dx ð5Þ
a !0þ a
ð 1 sin x
Note:Be alert to the word unbounded. This is distinct from undefined. For example, dx ¼
ð 1 sin x sin x 0 x
lim dx is a proper integral, since lim ¼ 1 and hence is bounded as x ! 0 even though the
!0 x x!0 x
function is undefined at x ¼ 0. In such case the integral on the left of (5)iscalled convergent or
divergent according as the limit on the right exists or does not exist.

Finally, the category of improper integrals of the second kind also includes the case where f ðxÞ
becomes unbounded only at an interior point x ¼ x 0 of the interval a @ x @ b, then we define
ð b ð x 0 1 ð b
f ðxÞ dx ¼ lim f ðxÞ dx þ lim f ðxÞ dx ð6Þ
a 1 !0þ a 2 !0þ x 0 þ 2
The integral on the left of (6)converges or diverges according as the limits on the right exist or do
not exist.
Extensions of these definitions can be made in case f ðxÞ becomes unbounded at two or more points
of the interval a @ x @ b.

CAUCHY PRINCIPAL VALUE
It may happen that the limits on the right of (6)do not exist when 1 and 2 approach zero
independently. In such case it is possible that by choosing 1 ¼ 2 ¼ in (6), i.e., writing
b b
ð ð x 0 ð
f ðxÞ dx ¼ lim f ðxÞ dx þ f ðxÞ dx ð7Þ
a !0þ a x 0 þ
the limit does exist. If the limit on the right of (7) does exist, we call this limiting value the Cauchy
principal value of the integral on the left. See Problem 12.14.

EXAMPLE. The natural logarithm (i.e., base e) may be defined as follows:
x dt
ð
;
ln x ¼ 0 < x < 1
1 t
1
is unbounded as x ! 0, this is an improper integral of the second kind (see Fig. 12-2). Also,
Since f ðxÞ¼
ð x
1 dt
is an integral of the third kind, since the interval to the right is unbounded.
0 t ð 1 dt
Now lim ¼ lim ½ln 1 ln Š! 1 as ! 0; therefore, this improper integral of the second kind is
!0 t !0
ð ð x
1 dt dt
divergent. Also, ¼ lim ¼ lim ½ln x ln iŠ! 1;thisintegral (which is of the first kind) also diverges.
1 t x!1 1 t x!1