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" Now S s S S S s " s and it follows that each term in parentheses is positive and so is less s " " than  by (4). In particular, since S s is a de nite number it must be zero, i.e., S s. Thus, the limits of the upper and lower sums are equal and the proof is complete.
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Supplementary Problems
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DEFINITION OF A DEFINITE INTEGRAL 1 5.32. (a) Express x3 dx as a limit of a sum.
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(b) Use the result of (a) to evaluate the given de nite integral.
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(c) Interpret the result geometrically. Ans. (b) 1 4 2 Using the de nition, evaluate (a) 3x 1 dx; 0 Ans. (a) 8, (b) 9
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x2 4x dx.
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& 5.34. Prove that lim
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INTEGRALS ' n n n  2 2 . 4 n2 12 n 22 n n2
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[CHAP. 5
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Prove that lim
5.36. 5.37.
' 1p 2p 3p np 1 if p > 1. n!1 p 1 np 1 b Using the de nition, prove that ex dx eb ea .
Work Problem 5.5 directly, using Problem 1.94 of 1. ) p 1 1 1 Prove that lim p p p ln 1 2 . n!1 2 12 2 22 2 n2 n n n
Prove that lim
n tan 1 x if x 6 0. 2 2 x n k x k 1
PROPERTIES OF DEFINITE INTEGRALS 5.40. Prove (a) Property 2, (b) Property 3 on Pages 91 and 92. b
If f x is integrable in a; c and c; b , prove that
f x dx
f x dx b
f x dx. b g x dx.
If f x and g x are integrable in a; b and f x @ g x , prove that
f x dx @
Prove that 1 cos x A x2 = for 0 @ x @ =2.  1   cos nx  dx @ ln 2 for all n. Prove that   x 1 
 p   3 e x sin x     Prove that  . dx @  1 x2 1  12e
MEAN VALUE THEOREMS FOR INTEGRALS 5.46. Prove the result (5), Page 92. [Hint: If m @ f x @ M, then mg x @ f x g x @ Mg x . Now integrate b and divide by g x dx. Then apply Theorem 9 in 3.
Prove that there exist values 1 and 2 in 0 @ x @ 1 such that 1 sin x 2  dx sin 2 2 2  1 1 4 0x 1 Hint: Apply the rst mean value theorem. 
(a) Prove that there is a value  in 0 @ x @  such that
e x cos x dx sin . (b) Suppose a wedge in the
shape of a right triangle is idealized by the region bound by the x-axis, f x x, and x L. Let the weight distribution for the wedge be de ned by W x x2 1. Use the generalized mean value theorem to show 3L L2 2 that the point at which the weighted value occurs is . 4 L2 3
CHAP. 5]
INTEGRALS
CHANGE OF VARIABLES AND SPECIAL METHODS OF INTEGRATION p 1 3 3 tan 1 t dx csch2 u p du, p ; d dt; c 5.49. Evaluate: (a) x2 esin x cos x3 dx; b 2 u 4x x2 0 1 t 1 2 dx (e) . 2 2 16 x 3 p Ans. (a) 1 esin x c; b 2 =32; c =3; d 2 coth u c; e 1 ln 3. 3 4 1 5.50. Show that (a) p 3 dx ; 2 3=2 12 0 3 2x x b dx p x2 x2 1 p x2 1 c. x
5.52. 5.53. 5.54.
p p p u2 a2 du 1 u u2 a2 1 a2 ln ju u2 a2 j Prove that (a) 2 2 p p (b) a2 u2 du 1 u a2 u2 1 a2 sin 1 u=a c; a > 0. 2 2 p p x dx x2 2x 5 ln jx 1 x2 2x 5j c. Find p : Ans. 2 2x 5 x Establish the validity of the method of integration by parts.  Evaluate (a)
x cos 3x dx; b
x3 e 2x dx:
Ans.
(a) 2=9;
b 1 e 2x 4x3 6x2 6x 3 c 3
1 1 1 x2 tan 1 x dx  ln 2 12 6 6 p p p   2 p 5 7 3 3 3 5 2 7 2 x 1 dx x b ln p . 4 4 8 2 3 3 2 Show that (a)
(a) If u f x and v g x have continuous nth derivatives, prove that uv n dx uv n 1 u 0 v n 2 u 00 v n 3 1 n u n v dx called generalized integration by parts. (b) What simpli cations occur if u n 0 Discuss. (c) Use (a) to
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