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1 12.62.
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1 12.63.
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1 12.64.
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IMPROPER INTEGRALS
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!   cos ax cos bx 1 2 b2 12.65. (a) Prove that e ; dx ln 2 x 2 a2 0   1 cos ax cos bx b dx ln . (b) Use (a) to prove that x a 0
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1 F ax F bx dx The results of (b) and Problem 12.60 are special cases of Frullani s integral, x 0!   1 b F t dt converges. F 0 ln , where F t is continuous for t > 0, F 0 0 exists and a t 1 1 e x dx 1 2
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p = , > 0. Prove that for p 1; 2; 3; . . ., p 1 2  1 3 5 2p 1 x2p e x dx 2p 1 =2 2 2 2 2 2 0 1 e a=x e b=x dx
12.67. If a > 0; b > 0, prove that
p p b a.
1 12.68. Prove that
  tan 1 x=a tan 1 x=b  b dx ln where a > 0; b > 0. x 2 a dx 4 p . x2 x 1 3 3 3 [Hint: Use Problem 12.38.]
1 12.69. Prove that
MISCELLANEOUS PROBLEMS ' 1& ln 1 x 2 dx converges. 12.70. Prove that x 0 " 1 X n 1  dx dx 12.71. Prove that converges. Hint: Consider and use the fact that 2 3 1 x3 sin2 x 0 1 x sin x n n 0 ! n 1  n 1  dx dx @ : 1 x3 sin2 x 1 n 3 sin2 x n n 1 1 12.72. Prove that
x dx diverges. 1 x3 sin2 x 1
12.73. (a) Prove that
ln 1 2 x2 dx  ln 1 ; A 0. 1 x2 0 =2  ln sin  d ln 2: (b) Use (a) to show that 2 0 1 sin4 x  dx . 3 x4 b lfcosh axg;
12.74. Prove that
12.75. Evaluate Ans:
c lf sin x =xg.   p s 1 ; s > 0: ; s > jaj c tan 1 a =s; s > 0 b 2 s s a2 b Evaluate lfeax sin bxg.
p (a) lf1= xg;
12.76. (a) If lfF x g f s , prove that lfeax F x g f s a ; Ans: b b ; s>a s a 2 b2
IMPROPER INTEGRALS
[CHAP. 12
12.77. (a) If lfF x g f s , prove that lfxn F x g 1 n f n s , giving suitable restrictions on F x . (b) Evaluate lfx cos xg. Ans: b s2 1 ; s>0 s2 1 2
12.78. Prove that l 1 ff s g s g l 1 f f s g l 1 fg s g, stating any restrictions. 12.79. Solve using Laplace transforms, the following di erential equations subject to the given conditions. (a) Y 00 x 3Y 0 x 2Y x 0; Y 0 3; Y 0 0 0 (b) Y 00 x Y 0 x x; Y 0 2; Y 0 0 3 (c) Y 00 x 2Y 0 x 2Y x 4; Y 0 0; Y 0 0 0 Ans. a Y x 6e x 3e 2x ; b Y x 4 2ex 1 x2 x; c Y x 1 e x sin x cos x 2 12.80. Prove that lfF x g exists if F x is piecewise continuous in every nite interval 0; b where b > 0 and if F x is of exponential order as x ! 1, i.e., there exists a constant such that je x F x j < P (a constant) for all x > b. 12.81. If f s lfF x g and g s lfG x g, prove that f s g s lfH x g where x H x F u G x u du
is called the convolution of F and G, written F G. & M '& M ' e su F u du e sv G v dv Hint: Write f s g s lim
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