Example 3 1 x2 Show that sin x is absolutely convergent. dx Solution By cauchy’s inequality, 2 dx x sin x x 1 1 2 1 dx dx x 1 dx sin 1 x x sin x 2 since x is positive and 1 (1) 86 b 1 = b = lim dx b 1 2 x Now lim b b x 1 1 1 lim b = 1 1 1 1 lim = b b =1 sin from (1) 1 x x dx 1 2 sin x dx Then is absolutely convergent. 1 x 2 Example 4 Show that xe dx is convergent. x 0 Solution
Example 3 Show that 1 sin x x2 dx is absolutely convergent. Solution By cauchy’s inequality, 1 sin x x dx 1 1 1 x sin x x 2 1 dx x 2 2 dx dx since x is positive and sin x 1 (1) 86 Now lim b x 1 2 dx = = b b 1 lim b 1 x b 1 = lim 1 b 1 1 = lim b 1 1 b =1 from (1) 1 sin x 2 x dx 1 Then 1 sin x x 2 dx is absolutely convergent. Example 4 Show that 0 xe x dx is convergent. Solution