Page 320 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 311

Fig. 12-2

SPECIAL IMPROPER INTEGRALS OF THE SECOND KIND
ð b dx
1. p converges if p < 1 and diverges if p A 1.
a ðx aÞ
ð b dx
2. p converges if p < 1 and diverges if p A 1.
a ðb xÞ
These can be called p integrals of the second kind. Note that when p @ 0 the integrals are proper.

CONVERGENCE TESTS FOR IMPROPER INTEGRALS OF THE SECOND KIND
The following tests are given for the case where f ðxÞ is unbounded only at x ¼ a in the interval
a @ x @ b. Similar tests are available if f ðxÞ is unbounded at x ¼ b or at x ¼ x 0 where a < x 0 < b.

1. Comparison test for integrals with non-negative integrands.
ð b
(a) Convergence. Let gðxÞ A 0 for a < x @ b, and suppose that gðxÞ dx converges. Then if
ð b a
0 @ f ðxÞ @ gðxÞ for a < x @ b, f ðxÞ dx also converges.
a
1 1 ð 5 dx
EXAMPLE. p ffiffiffiffiffiffiffiffiffiffiffiffiffi < p ffiffiffiffiffiffiffiffiffiffiffi for x > 1. Then since p ffiffiffiffiffiffiffiffiffiffiffi converges ( p integral with a ¼ 1,
4
x 1 x 1 1 x 1
ð 5 dx
1
p ¼ ), ffiffiffiffiffiffiffiffiffiffiffiffiffi also converges.
2 p 4
1 x 1
b
ð
(b) Divergence. Let gðxÞ A 0 for a < x @ b, and suppose that gðxÞ dx diverges. Then if
ð b a
f ðxÞ A gðxÞ for a < x @ b, f ðxÞ dx also diverges.
a
ln x 1 ð 6 dx
EXAMPLE. > for x > 3. Then since diverges ( p integral with a ¼ 3, p ¼ 4),
4 4 4
ðx 3Þ ðx 3Þ 3 ðx 3Þ
6 ln x
ð
dx also diverges.
4
3 ðx 3Þ
2. Quotient test for integrals with non-negative integrands.
ð b
(a)If f ðxÞ A 0 and gðxÞ A 0 for a < x @ b,and if lim f ðxÞ ¼ A 6¼ 0or 1, then f ðxÞ dx and
b
ð x!a gðxÞ a
gðxÞ dx either both converge or both diverge.
a