Let f (x) = sin x/x and I = 0 f (x) dx. We define f (0) = 1. Then f is continuous and I
Chapter 7, Problem 100(choose chapter or problem)
Let f (x) = sin x/x and I = 0 f (x) dx. We define f (0) = 1. Then f is continuous and I is not improper at x = 0. (a) Show that R 1 sin x x dx = cos x x R 1 R 1 cos x x2 dx (b) Show that 1 (cos x/x2)dx converges. Conclude that the limit as R of the integral in (a) exists and is finite. (c) Show that I converges. It is known that I = 2 . However, I is not absolutely convergent. The convergence depends on cancellation, as shown in Figure 13. y =
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