Page 325 - Schaum's Outline of Theory and Problems of Advanced Calculus

316 IMPROPER INTEGRALS [CHAP. 12

a s
fsin atg¼ 2 2 ; fcos atg¼ 2 2
s þ a s þ a
and recall that
00
f ðsÞ¼ fFðtÞg fF ðtÞg þ 4 fFðtÞg ¼ 3 fsin tg
Using (b)we obtain
3
2
s þ 1
s f ðsÞ s þ 4f ðsÞ¼ 2 :
Solving for f ðsÞ yields
3 s 1 1 s
:
2 2 2 2 2 2
f ðsÞ¼ þ ¼ þ
ðs þ 4Þðs þ 1Þ s þ 4 s þ 1 s þ 4 s þ 4
(Partial fractions were employed.)
Referring to the table of Laplace transforms, we see that this last expression may be written
1
2
f ðsÞ¼ fsin tg fsin 2tgþ fcos 2tg
then using the linearity of the Laplace transform
1
f ðsÞ¼ fsin t sin 2t þ cos 2tg:
2
We find that
1
FðtÞ¼ sin t sin 2t þ cos 2t
2
satisfies the differential equation.

IMPROPER MULTIPLE INTEGRALS
The definitions and results for improper single integrals can be extended to improper multiple
integrals.

Solved Problems

IMPROPER INTEGRALS
12.1. Classify according to the type of improper integral.
ð 1 dx ð 10 xdx ð 1 cos x
(a) p ffiffiffi (c) (e) dx
3 2 x 2
1 xðx þ 1Þ 3 ðx 2Þ 0
ð ð 2
dx x dx
1 1
(b) (d)
2
4
0 1 þ tan x 1 x þ x þ 1
(a) Second kind (integrand is unbounded at x ¼ 0 and x ¼ 1).
(b) Third kind (integration limit is infinite and integrand is unbounded where tan x ¼ 1Þ.
(c) This is a proper integral (integrand becomes unbounded at x ¼ 2, but this is outside the range of
integration 3 @ x @ 10).
(d) First kind (integration limits are infinite but integrand is bounded).