Page 399 - Schaum's Outline of Theory and Problems of Advanced Calculus

390 GAMMA AND BETA FUNCTIONS [CHAP. 15

ð =2 p
p
15.59. If 0 < p < 1prove that tan d ¼ sec .
0 2 2
ð 1 x u 1 v 1
15.60. Prove that ð1 xÞ Bðu; vÞ where u; v, and r are positive constants.
uþv ¼ u uþv
0 ðx þ rÞ r ð1 þ rÞ
[Hint: Let x ¼ðr þ 1Þy=ðr þ yÞ.]
ð =2 sin 2u 1 cos 2v 1 d
15.61. Prove that 2 uþv ¼ Bðu; vÞ where u; v > 0.
v u
2
0 ða sin þ b cos Þ 2a b
2
[Hint: Let x ¼ sin in Problem 15.60 and choose r appropriately.]
1 dx 1 1 1
ð
15.62. Prove that ¼ þ þ þ
0 x x 1 1 2 2 3 3
15.63. Prove that for m ¼ 2; 3; 4; ...
2 3 ðm 1Þ m
sin sin sin sin ¼
m m m m 2 m 1
m
[Hint: Use the factored form x 1 ¼ðx 1Þðx 1 Þðx 2 Þ ðx n 1 Þ,divide both sides by x 1, and
consider the limit as x ! 1.]

ð =2
15.64. Prove that ln sin xdx ¼ =2ln 2 using Problem 15.63.
0
[Hint: Take logarithms of the result in Problem 15.63 and write the limit as m !1 as a definite integral.]
ðm 1Þ=2
1 2 3 m 1
15.65. Prove that ¼ ð2 Þ p ffiffiffiffi :
m m m m m
[Hint: Square the left hand side and use Problem 15.63 and equation (11a), Page 378.]

ð 1
1
15.66. Prove that ln ðxÞ dx ¼ lnð2 Þ.
2
0
[Hint: Take logarithms of the result in Problem 15.65 and let m !1.]
ð
1 sin x
15.67. (a)Prove that p dx ¼ ; 0 < p < 1.
0 x 2 ð pÞ sinð p =2Þ
(b)Discuss the cases p ¼ 0 and p ¼ 1.
ð ð
1 1
3
2
15.68. Evaluate (a) sin x dx; ðbÞ x cos x dx.
0 0

p
Ans: ðaÞ 1 2 =2; ðbÞ p ffiffiffi
ffiffiffiffiffiffiffiffi
3 3 ð1=3Þ
x ln x
ð p 1
1
2
15.69. Prove that dx ¼ csc p cot p ; 0 < p < 1.
0 1 þ x
ð 2 p ffiffiffi
1 ln x 2
15.70. Show that dx ¼ .
4
0 x þ 1 16
2
15.71. If a > 0; b > 0, and 4ac > b ,prove that
ð ð
2 2
1 1 2
e ðax þbxyþcy Þ dx dy ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4ac b 2
1 1