Answers & Hints for Selected Exercises 685EXERCISE SET 7.8-A- Suppose f integrable on every [c, b] such that a < c < b. Suppose for contra-
diction that Ve > 0, f is bounded on [a, a+ 1::). Then, since f is also bounded
on [a+ 1::, b], it is bounded on [a, b]. By Thm. 7.4.7, f is integrable on [a, b].
Contradiction. - (a) Improper; diverges to +oo. (b) Improper; converges to 2.
(c) Not improper. (d) Improper; diverges to +oo.
( e) Improper; converges to -1. ( f) Improper; converges to 2J8.
(g) Improper; diverges. f 0
1
f diverges to +oo, f~ 1 f div. to -oo.
(h) Improper; diverges to +oo. (i) Improper; diverges to +oo.
(j) Not improper. (k) Improper; converges to 2(e - 1). (1) Not improper.
(m) Not improper. (n) Improper; diverges to -oo.
1
1;2 dx [-l] 1/2 11 dx
- (l ) 2 = lim - 1 - = 1/ In 2, converges. (I ) 2 =
0 x n x c-+O+ n x c^1 ;^2 x n x
lim [ 1 -^1 ]~ 12 = +oo, diverges. Therefore f
1
(ldx ) 2 diverges.
c-+ 1 - n x } o X n XEXERCISE SET 7.8-B- (a) Converges to ~· (b) Conv. to 1. (c) Diverges to +oo.
(d) Conv. to 2. (e) Div. to +oo. (f) Diverges. (g) Conv. to i·
(h) Conv. to 1/e. (i) Converges to 1r. (j) Conv. to 2 - 1/e.
(k) Conv. to 1. (1),(m),(n) Div. to +oo. - (a) Converges¢? r > 1. (b) Converges¢? r < 1. (c) Diverges for all r.
- (a) Conv. by comp. with ft" x-^312 dx. (b) Div. by comp. with Ji°'" x-^112 dx.
( c) Diverges, since Ji°'" f diverges by comparison with Ji'::o x-^112 dx.
(d) Conv. by comp. with ft° x-^312 dx. (e) Conv. by comp. with J~ (x!~l2.
(f) Converges, since J~! f conv. by comp. with J~! (x!~)2, J~ 3
1
f exists
since f is continuous on [-3, -1], and J~ f conv. by (e).- (a) For x ~ n, 0:::; lc~~xl :::; ~'and ft° ~dx converges.
(b) J b smxdx · = [ --^1 COSX J b - J b cosxdx SO Joo smxdx · = -1-Joo cosxdx.
1T x x 7r 7r"X2 ' 7r x 'Tr "X2
(c) { n, 2n, 3n, · · · , n + nn} is a partition of [n, (n + l)n }. Apply Cor. 7.4.3.(d) For even n , x E [nn, (n + l)n] =? sinx ~ ~ =? si~x ~ 2 (n!1)11";
D ror o dd n, x E [ nn, ( n + 1) 7r ] =? smx · < _ -1 2 =? sinx -x-< _ 2 (n+l)7r, -1.