306 CHAPTER 8 • RESIDUE THEORY. 2 (}
5. fi2" Sill dO.
(^0) 5 + 4cos8
. 2 (}
6. r2" Sill d(J.
Jo 5 - 3cos8
r2w 1- Jo 2 dO.
(5+ 3cos1J)
8 Ji2"1
dlJ-^0 (5+ 4cos0)^2 ·
9. fi2" cos 28 dB.
(^0 5) + 3cos8
lO. r2" cos W dO.
J o 13 - 12cos8
r2.,,. 1
11. Jo 2d0·
(1 + 3 cos^2 B)
12. Ji2"
1
0 2 dO.
(1+8cos^2 0)
13 f2" cos
(^2) 31J dO
·Jo 5 - 4cos28 ·
14. J.2" cos
2
38 dO.
(^0) 5-3cos28
15 J.
2
"
1
di}
-^0 acosO+ bsinO+ d '
where a, b, and d are re al and a^2 + b^2 < d^2 •J.2w 1- 0 2 (}. 2 d8, where a, b, and dare real and a> d and b > d.
acos +bsm O+d
8.3 Improper Integrals of Rational Funct ions
An important application of the theory of residues is the evaluation of certain
types of improper integrals. We let f be a continuous function of the real variable
x on the interval 0 :::; x < oo. Recall from calculus that the improper integral f
over [O,oo) is defined by(°" f (x) dx = Jim r f (x) dx,
Jo b-oo} 0