Prove that if ? is integrable on [?, ?] and ?(?) ≥ 0 for all ? in [?, ?], then
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Suppose that f is integrable on (a, b) and define (f(x) if f(x) > 0 f+(x) = 3 and f (2)= if f(x) < 0, Show that f+ and f- are integrable on (a, b), and If(x) if f(x) > 0, if f(x) < 0. cb Sisleyde = [* p*(e) ds + [°r(a)di. | f(x) dx = | f+(x) dx + 1 f (x) dx.
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
Problem 5. Let a < b and c > 0 and let f be integrable on [ca, cb]. Show that f c Ca where g(a) f(ex)
Exercise 6. Show that if f(x) > 0 for all x e [a, b] and f is integrable, then Sfdx > 0.
2.21 Let Q(2) = VI, which is defined for all x > 0. Prove: Q E C[0,0). (Hint: If a > 0, and € > 0, we seek 6 >0 such that 3 > 0 and - al <& implies Q(x) - Q(a) < €. Begin by showing that|vx-Val' <lt - al.)
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then
Prove that is an integer for all n > 0.
4. (a) Suppose that limz-c f(x) = L > 0. Prove that there exists a 8 >0 such that if 0 < 12 – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on (a, b) and f(x) > 0 for all x € (a, b), then there exists an e > 0 such that f(x) > € for all x € [a, b].
5. Prove that U(2") (n > 3) is not cyclic.
please prove part (b) use complex analysis and calculus of residue -dx neif a> 0 5. (a) x2+1 (b) For any real number a > 0, cos x dx ne"/a. a Hint: This is the real part of the integral obtained by replacing cos x by e