8.4 • IMPROPER INTEGRALS INVOLVING TRIGONOMETRIC FUNCTIONS 3117 J"° x2da:- -oo x^4 + 4 ·
8 Joo x
2
dx- -oo (x2 + 4 )3 ·
(^9) • Joo -oo (x2 + 1)2 dx (x2 + 4).
10 J"° x+2 dx
- -^00 (x2 + 4)(x^2 + 9) ·
oo 3x^2 +2- Loo (x2 + 4) (x2 + 9) dx.
(^12). J- oo^00 ~ x6 + i ·
13. Joo x
4
dx.
-oo xs + l
(^14) · Joo - oo (x2 + dx
0 2) (x2 + b2) •
where a > 0 and b > O.
15 Joo x
2
dx
- -oo (x2 + 0 2)3' where a> 0.
8.4 Improper Integrals Involving Trigonometric Functions
Let P and Q be polynomials of degree m and n, respectively, where n ~ m + l.
We can show (but omit the proof) that if Q (x) #- 0 for all real x, then
r {^00 P(x) [^00 P(x).P.. 1-oo Q(x) cosx dx and P.V. l-oo Q(x) smx dx
are convergent improper integrals. You may encounter integrals of this type in
the study of Fourier transforms and Fourier integrals. We now show how to
evaluate them.
Particularly important is our use of the identitiescos (ax) = Re (exp ( iax) J and sin (ax) = Im (exp ( iax)] ,
where a is a positive real number. The crucial step in the proof of Theorem 8.4
wouldn't hold if we were to use cos ( az) and sin ( az) instead of exp ( iaz), as you
will see when you get to Lemma 8.1.