Exercise 9 1. Classify the following into Improper integrals of the first kind and Improper integrals of the second kind. 1 sin  x x b). 2 x dx dx a). x  0 1 x dx x 3 dx c). d). 2 2 ( 3 ) x x 4 3 5 2. When will you say that the improper integral ( ) f x dx converges or a diverges. dx 3. When will you say that the improper integral converges or diverges. 2 x 3 2 92 b 4. Define the Cauchy’s principal value of the improper integral ( ) f x dx if f a (x) has a singularity at c a < c < b 5. Define absolute convergence and conditional convergence of an improper b integral ( ) f x dx a dx 6. Show that is divergent . 0 x 5 dx 7. Show that is divergent. 1 x 5 1 is absolutely convergent. cos3xdx 8. Show that e x 0 x dx 2 is absolutely convergent or not. 1
Exercise 9 1. Classify the following into Improper integrals of the first kind and Improper integrals of the second kind. a). 0 x 2  1 x 1  dx b). x sin x x dx c). 3 2 dx x 4 d). 5 2 x 3 ( x 3 ) dx 2. When will you say that the improper integral a f ( x ) dx converges or diverges. 3. When will you say that the improper integral 2 2 dx x 3 converges or diverges. 92 4. Define the Cauchy’s principal value of the improper integral b a f ( x ) dx if f (x) has a singularity at c a < c < b 5. Define absolute convergence and conditional convergence of an improper integral b a f ( x ) dx 6. Show that 7. Show that 1 0 1 dx x 5 dx x 5 is divergent . is divergent. 8. Show that e 0 x cos3xdx is absolutely convergent. x 2 1 dx is absolutely convergent or not.
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