Page 121 - Schaum's Outline of Theory and Problems of Advanced Calculus

112 INTEGRALS [CHAP. 5

n n n
5.34. Prove that lim þ þ þ ¼ .
2
2
2
n!1 n þ 1 2 n þ 2 2 n þ n 2 4
p p p p
1 þ 2 þ 3 þ þ n 1
5.35. Prove that lim ¼ if p > 1.
n pþ1 p þ 1
n!1
ð b
a
x
b
5.36. Using the definition, prove that e dx ¼ e e .
a
5.37. Work Problem 5.5 directly, using Problem 1.94 of Chapter 1.
( )
1 1 1 p ffiffiffi
5.38. Prove that lim p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ lnð1 þ 2Þ.
2
2
2
n!1 n þ 1 2 n þ 2 2 n þ n 2
n 1
X n tan x
5.39. Prove that lim ¼ if x 6¼ 0.
2
2 2
n þ k x x
k¼1
n!1
PROPERTIES OF DEFINITE INTEGRALS
5.40. Prove (a)Property 2, (b)Property 3 on Pages 91 and 92.
ð b ð c ð b
5.41. If f ðxÞ is integrable in ða; cÞ and ðc; bÞ,prove that f ðxÞ dx ¼ f ðxÞ dx þ f ðxÞ dx.
a a c
ð b ð b
5.42. If f ðxÞ and gðxÞ are integrable in ½a; bŠ and f ðxÞ @ gðxÞ,prove that f ðxÞ dx @ gðxÞ dx.
a a
2
5.43. Prove that 1 cos x A x = for 0 @ x @ =2.
1
ð
cos nx
5.44. Prove that dx @ ln 2 for all n.

0 x þ 1

p ffiffi
3 x sin x
e
ð
5.45. Prove that 2 dx @ .

1 x þ 1 12e
MEAN VALUE THEOREMS FOR INTEGRALS
5.46. Prove the result (5), Page 92. [Hint: If m @ f ðxÞ @ M,then mgðxÞ @ f ðxÞgðxÞ @ MgðxÞ. Now integrate
ð b
and divide by gðxÞ dx. Then apply Theorem 9 in Chapter 3.
a
5.47. Prove that there exist values 1 and 2 in 0 @ x @ 1suchthat
ð 1
sin x 2
2
2
0 x þ 1 dx ¼ ð þ 1Þ ¼ 4 sin 2
1
Hint: Apply the first mean value theorem.
ð
5.48. (a)Prove that there is a value in 0 @ x @ such that e x cos xdx ¼ sin .(b) Suppose a wedge in the
0
shape of a right triangle is idealized by the region bound by the x-axis, f ðxÞ¼ x, and x ¼ L.Let the weight
2
distribution for the wedge be defined by WðxÞ¼ x þ 1. Use the generalized mean value theorem to show
2
3L L þ 2
that the point at which the weighted value occurs is .
2
4 L þ 3