Page 124 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 5] INTEGRALS 115

dx sin 2x dx
ð ð =2 ð
1 1
5.76. Evaluate (a) 3 ; ðbÞ 4=3 dx; ðcÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffi.
0 1 þ x 0 ðsin xÞ x þ 1
2
2 0 x þ
Ans. (a) p ffiffiffi ðbÞ 3 ðcÞ does not exist
3 3
2 Ð =2 sin t dt
e
5.77. Evaluate lim ex = e =4 þ x . Ans. e=2
x! =2 1 þ cos 2x
ð 3 ð 2
x
x
d d
2
3
5
3
2
2
4
2
5.78. Prove: (a) ðt þ t þ 1Þ dt ¼ 3x þ x 2x þ 3x 2x; ðb cos t dt ¼ 2x cos x cos x .
dx x 2 dx x
ð ð =2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx p ffiffiffi p ffiffiffi
p
5.79. Prove that (a) 1 þ sin x dx ¼ 4; ðbÞ ¼ 2 lnð 2 þ 1Þ.
0 0 sin x þ cos x
ð 1 dx ð 1 dy
5.80. Explain the fallacy: I ¼ ¼ ¼ I,using the transformation x ¼ 1=y.Hence I ¼ 0.
1 1 þ x 2 1 1 þ y 2
1
1
But I ¼ tan ð1Þ tan ð 1Þ¼ =4 ð =4Þ¼ =2. Thus =2 ¼ 0.
1=2 cos x 1 1 1
ð
5.81. Prove that p ffiffiffiffiffiffiffiffiffiffiffiffiffi dx @ tan .
0 1 þ x 2 4 2
( ffiffiffiffiffiffiffiffiffiffiffiffiffiffi )
p ffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffi p
2n 1 p ffiffiffi
5.82. Evaluate lim n þ 1 þ n þ 2 þ þ . Ans. 2 3 ð2 2 1Þ
n 3=2
n!1

1if x is irrational
5.83. is not Riemann integrable in ½0; 1Š.
0if x is rational
Prove that f ðxÞ¼
[Hint: In (2), Page 91, let k , k ¼ 1; 2; 3; ... ; n be first rational and then irrational points of subdivision and
examine the lower and upper sums of Problem 5.31.]
5.84. Prove the result (3)ofProblem 5.31. [Hint: First consider the effect of only one additional point of
subdivision.]
s
5.85. In Problem 5.31, prove that s @ S. [Hint: Assume the contrary and obtain a contradiction.]
ð b
5.86. If f ðxÞ is sectionally continuous in ½a; bŠ,prove that f ðxÞ dx exists. [Hint: Enclose each point of disconti-
a
nuity in an interval, noting that the sum of the lengths of such intervals can be made arbitrarily small. Then
consider the difference between the upper and lower sums.
8
2x 0 < x < 1 ð
< 2
5.87. If f ðxÞ¼ 3 x ¼ 1 , find f ðxÞ dx. Interpret the result graphically. Ans. 9
6x 11 < x < 2
: 0
3
ð
1
5.88. Evaluate fx ½xŠþ g dx where ½xŠ denotes the greatest integer less than or equal to x.Interpret the result
2
0
graphically. Ans.3
m
ð =2 sin x
5.89. (a)Prove that m m dx ¼ for all real values of m.
0 sin x þ cos x 4
ð 2 dx
(b)Prove that ¼ .
4
0 1 þ tan x
ð =2 sin x
5.90. Prove that dx exists.
0 x
ð 0:5 tan 1 x
5.91. Show that dx ¼ 0:4872 approximately.
0 x
ð xdx 2
5.92. Show that 2 ¼ p :
ffiffiffi
0 1 þ cos x 2 2