Page 342 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 333
!

ð 2 2
1 x cos ax cos bx 1 þ b
12.65. (a)Prove that e dx ¼ ln 2 2 ; A 0.
0 x 2 þ a

ð
1 cos ax cos bx b
(b)Use (a)toprove that dx ¼ ln .
0 x a
ð
1
The results of (b) and Problem 12.60 are special cases of Frullani’s integral, FðaxÞ FðbxÞ
x dx ¼
ð 0
b 1
Fð0Þ ln , where FðtÞ is continuous for t > 0, F ð0Þ exists and FðtÞ dt converges.
0
a 1 t
ð
1 2
12.66. Given e x 1 p ffiffiffiffiffiffiffiffi Prove that for p ¼ 1; 2; 3; ...,
= , > 0.
2
dx ¼
0
ð p ffiffiffi
1 2p x 2 1 3 5 ð2p 1Þ
x e dx ¼
0 2 2 2 2 2 ð2pþ1Þ=2
ð
1 2 2 p ffiffiffiffiffiffi p ffiffiffiffiffiffi
12.67. If a > 0; b > 0, prove that ðe a=x e b=x Þ dx ¼ b a.
0
ð 1 1
1 b
12.68. Prove that tan ðx=aÞ tan ðx=bÞ dx ¼ ln where a > 0; b > 0.
0 x 2 a
ð
1 dx 4
12.69. Prove that ¼ p . [Hint: Use Problem 12.38.]
2 3 3 3 ffiffiffi
1 ðx þ x þ 1Þ
MISCELLANEOUS PROBLEMS
2
ð
1
12.70. Prove that lnð1 þ xÞ dx converges.
0 x
"
ð 1 ð ðnþ1Þ
dx X dx
1
12.71. Prove that 2 converges. Hint: Consider 2 and use the fact that
3
3
0 1 þ x sin x n¼0 n 1 þ x sin x
ð ðnþ1Þ ð ðnþ1Þ
dx @ dx :
3
2
2
3
n 1 þ x sin x n 1 þðn Þ sin x
ð
1 xdx
12.72. Prove that diverges.
2
3
0 1 þ x sin x
ð 2 2
1
12.73. (a)Prove that lnð1 þ x Þ dx ¼ lnð1 þ Þ; A 0.
2
0 1 þ x
=2
ð
(b)Use (a)toshow that ln sin d ¼ ln 2:
0 2
ð 4
1 sin x
12.74. Prove that dx ¼ .
x 4 3
0
ffiffiffi
p
12.75. Evaluate (a) lf1= xg; ðbÞ lfcosh axg; ðcÞ lfðsin xÞ=xg.
s
p ffiffiffiffiffiffiffi 1 1
Ans: ðaÞ =s; s > 0 ðbÞ ; s > jajðcÞ tan ; s > 0:
2
s a 2 s
ax
ax
12.76. (a)If lfFðxÞg ¼ f ðsÞ,prove that lfe FðxÞg ¼ f ðs aÞ; ðbÞ Evaluate lfe sin bxg.
b
Ans: ðbÞ ; s > a
2
ðs aÞ þ b 2