Page 341 - Schaum's Outline of Theory and Problems of Advanced Calculus

332 IMPROPER INTEGRALS [CHAP. 12

ð ð p ffiffiffi
1 sin x 1 sin x
ffiffiffi dx; ffiffiffi dx: Ans: ðaÞ cond. conv., ðbÞ abs. conv.
p p
ðaÞ ðbÞ
0 x 0 sinh x
UNIFORM CONVERGENCE OF IMPROPER INTEGRALS
cos x
ð
1
0 1 þ x
12.50. (a)Prove that ð Þ¼ 2 dx is uniformly convergent for all .
(b)Prove that ð Þ is continuous for all . (c)Find lim ð Þ: Ans. (c) =2:
!0
ð
1 2
2
Fðx; Þ dx, where Fðx; Þ¼ xe x . (a) Show that ð Þ is not continuous at ¼ 0, i.e.,
12.51. Let ð Þ¼
0
ð ð
1 1
lim lim Fðx; Þ dx. (b) Explain the result in (a).
!0 0 Fðx; Þ dx 6¼ 0 !0
2
12.52. Work Problem 12.51 if Fðx; Þ¼ xe x .
12.53. If FðxÞ is bounded and continuous for 1 < x < 1 and
1 ð 1 yFð Þ d
Vðx; yÞ¼ 2
2
1 y þð xÞ
prove that lim Vðx; yÞ¼ FðxÞ.
y!0
12.54. Prove (a) Theorem 7 and (b) Theorem 8 on Page 314.
12.55. Prove the Weierstrass M test for uniform convergence of integrals.
ð ð
1 1
12.56. Prove that if FðxÞ dx converges, then e x FðxÞ dx converges uniformly for A 0.
0 0
ð
1 ax sin x 1
e dx converges uniformly for a A 0, tan a,
12.57. Prove that ðaÞ ðaÞ¼ ðbÞ ðaÞ¼
0 x 2
ð
1 sin x
(c) dx ¼ (compare Problems 12.27 through 12.29).
0 x 2
12.58. State the definition of uniform convergence for improper integrals of the second kind.
12.59. State and prove a theorem corresponding to Theorem 8, Page 314, if a is a differentiable function of .
EVALUATION OF DEFINITE INTEGRALS
Establish each of the following results. Justify all steps in each case.
ð 1 ax e bx
e
12.60. dx ¼ lnðb=aÞ; a; b > 0
0 x
e
ð 1 ax e bx
1
1
12.61. dx ¼ tan ðb=rÞ tan ða=rÞ; a; b; r > 0
0 x csc rx
sin rx
ð
1
r
12.62. ð1 e Þ; r A 0
2 dx ¼ 2
0 xð1 þ x Þ
1 cos rx
ð
1
12.63. dx ¼ jrj
x 2 2
0
ð
x sin rx ar
1
12.64. dx ¼ e ; a; r A 0
2
0 a þ x 2 2