Page 397 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 397
388 GAMMA AND BETA FUNCTIONS [CHAP. 15
ð ð ð
1 1 1 2
4 x
2 2x
6 3x
15.29. Evaluate (a) x e dx; ðbÞ x e dx; ðcÞ x e dx:
0 0 0
ffiffiffiffiffiffi
p
80 2
Ans. ðaÞ 24; ðbÞ ; ðcÞ
243 16
ð ð ð
1 2 1 p ffiffiffi p ffiffi 1 5
3 2y
15.30. Find (a) e x dx; ðbÞ 4 x e x dx; ðcÞ y e dy.
0 0 0
p
ffiffiffi
3
1
1 ð Þ; ; ð4=5Þ
3 3 2 5 16
Ans. ðaÞ ðbÞ ðcÞ p ffiffiffiffiffi
5
ffiffiffi
e
ð 1 st r
15.31. Show that p ffiffi dt ¼ ; s > 0.
0 t 8
1
ð 1 v 1
ln dx; v > 0.
15.32. Prove that ðvÞ¼ x
0
ð 1 ð 1 ð 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
4
15.33. Evaluate (a) ðln xÞ dx; ðbÞ ðx ln xÞ dx; ðcÞ 3 lnð1=xÞ dx.
0 0 0
1
Ans: ðaÞ 24; ðbÞ 3=128; ðcÞ 1 3 ð Þ
3
ffiffiffi
15.34. Evaluate (a) ð 7=2Þ; ðbÞ ð 1=3Þ. Ans: ðaÞð16 Þ=105; ðbÞ 3 ð2=3Þ
p
15.35. Prove that lim ðxÞ¼ 1 where m ¼ 0; 1; 2; 3; ...
x! m
m m
ffiffiffi
ð 1Þ 2 p
1
2
15.36. Prove that if m is a positive interger, ð m þ Þ¼
1 3 5 ð2m 1Þ
ð
1
e ln xdx is a negative number (it is equal to
, where
¼ 0:577215 ... is called
0 x
15.37. Prove that ð1Þ¼
0
Euler’s constant as in Problem 11.49, Page 296).
THE BETA FUNCTION
p
15.38. Evaluate (a) Bð3; 5Þ; ðbÞ Bð3=2; 2Þ; ðcÞ Bð1=3; 2=3Þ: Ans: ðaÞ 1=105; ðbÞ 4=15; ðcÞ 2 = 3 ffiffiffi
1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
ð ð ð
3
2 3=2
2
15.39. Find (a) x ð1 xÞ dx; ðbÞ ð1 xÞ=x dx; ðcÞ ð4 x Þ dx.
0 0 0
Ans: ðaÞ 1=60; ðbÞ =2; ðcÞ 3
ð 4 ð 3 dx
15.40. Evaluate (a) u 3=2 ð4 uÞ 5=2 du; ðbÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : Ans: ðaÞ 12 ; ðbÞ
0 0 3x x 2
a dy
ð 2
ffiffiffiffiffiffi :
15.41. Prove that p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ f ð1=4Þg
4
p
0 a y 4 4a 2
ð =2 ð 2
6
4
4
15.42. Evaluate (a) sin cos d ; ðbÞ cos d : Ans: ðaÞ 3 =256; ðbÞ 5 =8
0 0
ð ð =2
5
5
2
15.43. Evaluate (a) sin d ; ðbÞ cos sin d : Ans: ðaÞ 16=15; ðbÞ 8=105
0 0
ð
ffiffiffi
=2 p
ffiffiffiffiffiffiffiffiffiffi p
15.44. Prove that tan d ¼ = 2.
0
