Page 340 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 331

sin 2x
ð
1
12.40. Test for convergence, indicating absolute or conditional convergence where possible: (a) 3 dx;
0 x þ 1
ð ð ð
1 2 1 cos x 1 x sin x
(b) e ax cos bx dx, where a; b are positive constants; (c) p ffiffiffiffiffiffiffiffiffiffiffiffiffi dx; ðdÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx;
2
2
0 x þ 1 0 x þ a 2
1
ð
1 cos x
(e) dx.
0 cosh x
Ans. (a) abs. conv., (b) abs. conv., (c)cond. conv., (d)div., (e) abs. conv.
12.41. Prove the quotient tests (b) and (c)on Page 309.
IMPROPER INTEGRALS OF THE SECOND KIND
12.42. Test for convergence:
ð 1 dx ð 2 ln x ð 3 x 2 ð 1 dx
ffiffiffiffiffiffiffiffiffiffiffiffiffi dx dx
p ffiffiffiffiffiffiffiffiffiffiffiffiffi p
ðaÞ 2 ðdÞ 3 3 ðgÞ 2 ð jÞ x
0 ðx þ 1Þ 1 x 1 8 x 0 ð3 xÞ 0 x
1 cos x 1 dx e cos x
ð ð ð =2 x
dx dx
ðbÞ 2 ðeÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðhÞ
0 x 0 lnð1=xÞ 0 x
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
ð 1 tan x ð =2 ð 1 2 2
e 1 k x
dx ln sin xdx dx; jkj < 1
ðcÞ x ð f Þ ðiÞ 2
1 0 0 1 x
Ans. (a)conv., (b)div., (c)div., (d)conv., (e)conv., ( f Þ conv., (g)div., (h)div., (i)conv.,
( jÞ conv.
ð 5 dx
12.43. (a)Prove that diverges in the usual sense but converges in the Cauchy principal value senses.
0 4 x
(b)Find the Cauchy principal value of the integral in (a) and give a geometric interpretation.
Ans. (b)ln 4
12.44. Test for convergence, indicating absolute or conditional convergence where possible:
ð 1 ð 1 1 ð 1 1
1
1
1
cos dx; cos dx; cos dx:
ðaÞ x ðbÞ x ðcÞ 2 x
0 0 x 0 x
Ans. (a) abs. conv., (b)cond. conv., (c)div.
4
p
ð 1 1 32 2 ffiffiffi
2
12.45. Prove that 3x sin x cos dx ¼ 3 .
0 x x
IMPROPER INTEGRALS OF THE THIRD KIND
ð ð x ð x
1 1 e dx 1 e dx
12.46. Test for convergence: (a) e x ln xdx; ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; .
p
ðbÞ
p
ðcÞ
3 ffiffiffi
0 0 x lnðx þ 1Þ 0 x ð3 þ 2 sin xÞ
Ans. (a)conv., (b)div., (c)conv.
ð ð x
1 dx 1 e dx
12.47. Test for convergence: (a) p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðbÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; a > 0.
3 4 2
0 x þ x 0 sinh ðaxÞ
Ans. (a)conv., (b)conv. if a > 2, div. if 0 < a @ 2.
ð
1
12.48. Prove that sinh ðaxÞ dx converges if 0 @ jaj < and diverges if jaj @ .
0 sinh ð xÞ
12.49. Test for convergence, indicating absolute or conditional convergence where possible: