1549901369-Elements_of_Real_Analysis__Denlinger_

7.6 The Fundamental Theorem of Calculus 411 From this we would write rv:rr Jo xsin(x^2 +~)dx = ~ [-cosuJ!/~ 2 =~[-cos^3 ; +cos~] ...

412 Chapter 7 • The Riemann Integral a and x, where x -=/=-a. Suppose f(n+I) (t) exists and is integrable on [a, x ] if a < x ...

7.6 The Fundamental Theorem of Calculus 413 J(k+l)( ) = - (k + 1 ~ (x - a)k+I + [f(x) - Tk(x)] (fill in details) = f(x) -Tk+1(x) ...

414 Chapter 7 • The Riemann Integral Then Vx E [a, b], mg(x) :::; f(x)g(x) :::; Mg(x), so by Theorems 7.5.2 (d) and 7.5.l (a), ( ...

7.6 The Fundamental Theorem of Calculus 415 1 (b) \:/x E (0, 1), 0 < 'l/Jn(x) < 1. n. ( c) 'I/Jn ( 0) = 0 and the successi ...

416 Chapter 7 • The Riemann Integral Thus, by the fundamental theorem of calculus, 1 1 ePxp^2 n+l'l/Jn(x)dx = [ ePx Fn(x) t = eP ...

7 .6 The Fundamental Theorem of Calculus 417 It is straightforward to prove (Exercise 19) that F;:(x) = bnnZn+Z'l/Jn(x) -n^2 Fn( ...

418 Chapter 7 • The Riemann Integral have been extended and updated by Ivan Niven in his enlightening monograph, Irrational Numb ...

7.6 The Fundamental Theorem of Calculus 419 { x^2 sin (~) if x # 0 } In Chapter 6 we proved that the function f(x) = x 0 ifx=O ...

420 Chapter 7 • The Riemann Integral Find each of the following (see Exercise 11): d r (a) dx}o J3t^2 +5dt (c) -d^10 (2t^3 + 4 ...

7.6 The Fundamental Theorem of Calculus 421 Let F(x) =fox T, where T denotes Thomae's function. Prove that (a) Fis differentia ...

422 Chapter 7 • The Riemann Integral 7 *Elementary Transcendental Functions In this section we define the exponential,^15 loga ...

7 *Elementary 'Iranscendental Functions 423 Theorem 7.7.3 (Laws of Logarithms) \:/x, y E (0, +oo), and \:/n EN, (a) ln(xy) = l ...

424 Chapter 7 • The Riemann Integral In calculus courses we usually justify the existence of such a number by ap- pealing to the ...

7.7 *Elementary 'Ifanscendental Functions 425 (d) The range ofln: (0, +oo) --t JR is JR; the graph is as shown in Figure 7.11. y ...

426 Chapter 7 11 The Riemann Integral For those who skipped Section 5.6. The only previous definition we must be consistent with ...

7.7 *Elementary Transcendental Functions 427 Proof. Suppose f: JR ~ ( 0, oo) satisfies the differential equation f' = k f, Vx ER ...

428 Chapter 7 • The Riemann Integral (a) Vx E (O,oo), loga(ax) = x and a^10 g"x = x. (b) (c) If a > 1, the function loga x is ...

7.7 *Elementary Transcendental Functions 429 ( c) arcsin x is differentiable on ( -1, 1), and dd = ~. x l - x^2 (d) arcsinx is a ...

430 Chapter 7 • The Riemann Integral Definition 7.7.23 (of sinx, for All Real Numbers) (a) For~ :::; x:::;^3 ;, we define sin x ...