In other words b ( )dx f x is convergent if a is convergent. ( ) f x dx b a Example 5 Show that e cos 2x dx is absolutely convergent. x 0 Solution We must show that e cos 2x dx - x 0 is convergent. 88 0 lim cos 2x dx b = e - x b e cos 2x dx 0 - x b lim b 0 l 0 1 lim 1 b e b -1 0 lim b is convergent since the limit exists b e cos 2x dx - x - x e 0 is convergent. cos 2x dx x e Hence 0 is absolutely convergent. cos 2x 9.8 Convergence and divergence of the improper integrals of the second kind. b Consider the integral suppose that f(x) has a singularity at an ( x) dx f a a interior c of
In other words b a f ( x )dx is convergent if b a f ( x ) dx is convergent. Example 5 Show that e 0 x cos 2x dx is absolutely convergent. Solution We must show that 0 e - x cos 2x dx is convergent. 88 0 e - x cos 2x dx = lim b b 0 e - x cos 2x dx b lim b 0 l 0 lim b -1 1 e b 1 lim b b 0 e - x cos 2x dx is convergent since the limit exists e 0 - x cos 2x dx is convergent. Hence e 0 x cos 2x is absolutely convergent. 9.8 Convergence and divergence of the improper integrals of the second kind. Consider the integral b a f ( x) dx suppose that f(x) has a singularity at an interior c of a
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