Let f, : [a, ] R. Abel's Test asserts that if f R*[a, ]

Question:

Let f, φ : [a, ∞] → R. Abel's Test asserts that if f ∈ R*[a, ∞] and φ is bounded and monotone on [a, ∞], then f φ ∈ R*[a, ∞].
(a) Show that Abel's Test does not apply to establish the convergence of ∫∞0 (1/x)sin x dx by taking φ(x) := 1/x. However, it does apply if we take φ(x) := 1/√x and use Exercise 14.
(b) Use Abel's Test and Exercise 15 to show the convergence of ∫∞0(x/(x + 1)) sin(x2) dx.
(c) Use Abel's Test and Exercise 14 to show the convergence of ∫∞0 x-3/2(x + 1) sin x dx.
(d) Use Abel's Test to obtain the convergence of Exercise 16(f).