1549901369-Elements_of_Real_Analysis__Denlinger_

7.7 *Elementary Transcendental Functions 431 Apply Theorems 6.1.13 and 6.1.14. • Corollary 7. 7. 27 The sine function is differe ...

432 Chapter 7 • The Riemann Integral Theorem 7.7.31 The sine and cosine functions obey the following identities: \ix,yEJR, (a) s ...

7.7 *Elementary Transcendental Functions 433 Proof. Suppose F: JR~ JR has properties (a)-(c). By (b), F(O) = sinO. So, suppose x ...

434 Chapter 7 • The Riemann Integral Theorem 7.7.35 (Characterization of the Cosine Function) The only function F : JR ____, JR ...

7.8 *Improper Riemann Integrals 435 l: f an improper integral of type I. If lim _t exists (as a real number) c->b- a then we ...

436 Chapter 7 • The Riemann Integral Now, for 0 <: c < 1, 1 1 )xdx= [2v'x]~=2(1-y'c). Thus, lim j 1 ~ dx = 2, so f 1 ~ dx ...

7.8 *Improper Riemann Integrals 437 Proof. Suppose l: f is an improper integral of Type I. We consider the case in which f is in ...

438 Chapter 7 Iii The Riemann Integral (g)^11 -d^1 x^15 1 12 1 x3 (h)^2 (x - 3)2 dx (i) -2 --dx x - l -1 (j)^12 dx -1 x2 +ex (k) ...

7.8 *Improper Riemann Integrals 439 Definition 7.8.11 Suppose that 'ti a< b, f is integrable on [a, b]. Then we call J~;: f a ...

440 Chapter 7 • The Riemann Integral (Of course, if la+oo f diverges, then so does la+oo g.) Proof. Exercise 8. • Theorem 7.8.14 ...

7.8 *Improper Riemann Integrals 441 Solution. We consider two separate improper integrals: / + oo 1 r;;; dx. 1 yxex 1 1 1 r;;; d ...

442 Chapter 7 • The Riemann Integral (a) 1 (^00) dx 1 x3 1 (^00 1) (g) --dx o 1 + x2 J oo 1 (i) --dx -oo 1 + x2 roo x (m) 11 (x2 ...

7.9 *Lebesgue's Criterion for Riemann Integrability 443 (c) 1 (n+l)7r I sin xi n l(k+l)?r I sin x i Show that 'Vn EN, · --dx = L ...

444 Chapter 7 • The Riemann Integral Unfortunately, the proof is somewhat complicated. You may even decide the proof is indigest ...

7.9 *Lebesgue's Criterion for Riemann Integrability 445 Remarks 7.9.3 Let a > 0. (a) \:/nonempty A~ V(f), w1(A) 2'. 0. (b) \: ...

446 Chapter 7 • The Riemann Integral 8E: Putting (46) and (47) together, 5 I:; L.i < -. Therefore, iEN 2 c I:: L.i < -. • ...

7.9 *Lebesgue's Criterion for Riemann Integrability 447 The collection {Ux : x E [a, b]} is an open cover of [a, b], which is co ...

448 Chapter 7 • The Riemann Integral Let c > 0. To prove that f is integrable on [a, b], it suffices to prove that there is a ...

7.9 *Lebesgue's Criterion for Riemann Integrability 449 Now, L (Mi - mi)6.i ::; L (M - m)6.i iEN1 iEN1 = ( M - m) l: l (Ii) iEN1 ...

450 Chapter 7 • The Riemann Integral (e) Dirichlet's function, defined in Example 5.1.11, is not integrable on any proper interv ...