Pre-Calc Homework Solutions 340

Pre-Calc Homework Solutions 340 - 340 1 Section 8.3 1 36. 1...

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Unformatted text preview: 340 1 Section 8.3 1 36. 1 ln x dx 2 0 ln x dx by symmetry of ln x about the 40. dx 0 1 0 dx x b 1 dx x dx x b y-axis. Integrate ln x dx by parts. 1 x dx x lim b 1 u du ln x 1 dx x dv v x ln x 2 lim b0 b dx x lim 2 b x 1 lim (2 b b 2) ln x dx 1 dx 1 x ln x ln x dx 1 x C Since this integral diverges, the given integral diverges. 41. 0 1 x dx x cos x on [ , ) x b dx lim lim ln x x b b 2 2 0 ln x dx b 2 lim b0 b0 x ln x x b lim (ln b b ln ) 2 lim [ 1 Note that lim b ln b b0 b ln b lim b0 b] lim b0 2 1/b 1/b 2 Since this integral diverges, the given integral diverges. 42. 0 1 2 dx x2 sin x x2 2 on [ , x2 b ln b 1/b ) lim b0 b 0. lim b 2x 2x 2b 2 dx b The integral converges. 37. 0 1 1 on [1, ) 1 e e b 1 d lim e d e 1 b b lim b 1 lim b 1 2 2 1 lim b e 1 Since this integral converges, the given integral converges. e 1] 43. First rewrite 1 ex e x 1 ex e 1 lim [ e b b x. 1 e Since this integral converges, the given integral converges. 38. 0 1 x dx x b 1 x2 b on [2, ) 1 dx x b lim 2 lim ln x b 2 lim (ln b b ln 2) ex e x(e2x 1) 1 (e x)2 e x dx Integrate by letting u e x so du 1 (e x)2 dx e x dx x x e e 1 (e x)2 du 1 u2 e x dx. 2 tan tan 1 u ex dx e 0 C C x Since this integral diverges, the given integral diverges. 1 39. Let f (x) [1, ). x 1 and g(x) x2 1 . Both are continuous on x3/2 0 dx e dx ex e ex 0 x ex 0 ex dx e x x lim b b ex dx e 1 x x 1 f (x) lim lim x g(x) x x x b 1 dx lim x 3/2 dx x3/2 1 b 1 lim 1 1 x 1 lim b tan ex 1 0 b lim [tan b 1 tan 1 e b] lim b 2x 1/2 b 1 4 0 b lim ( 2b b 1/2 2) 2 0 dx ex e x lim b 0 4 dx ex e 1 x Since this integral converges, the given integral converges. lim tan b ex e b b 0 lim [tan b 1 tan 1 1] 2 4 4 Thus, the given integral converges. ...
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