 x l c c b Now = b ( x) dx f + ( x) ( x) dx a f dx a f (1) b The improper integral is said to be convergent if both the improper ( x) dx f a integrals on the right side of (1) exist. The sum of these is defined as the value of the improper integral from a to b in (1) 89 b 9.9 The Cauchy’s principal value of ( x) dx f a is a singular point of f(x). a c b If c b = + does not exist but ( x) dx f a ( x) dx ( x) dx f f lim ( x) dx b f does exist. We call this limit  0 a  c  b the Cauchy’s principal value of ( x) dx f a Thus the Cauchy’s principal value of b  c ( x) dx lim f (x)dx  f  0 a a b Example 6 1 is improper. dx a) Show that 1 x 2 2 b) Find the Cauchy’s principle value of 1
x l  Now b a f ( x) dx = c a f ( x) dx b + f ( x) c dx (1) The improper integral b a f ( x) dx is said to be convergent if both the improper integrals on the right side of (1) exist. The sum of these is defined as the value of the improper integral from a to b in (1) 89 b 9.9 The Cauchy’s principal value of f ( x) dx a is a singular point of f(x). If b a f ( x) dx = c a f ( x) dx + b c f ( x) dx does not exist but lim  0 a c   b f ( x) dx does exist. We call this limit the Cauchy’s principal value of b a f ( x) dx Thus the Cauchy’s principal value of b a f ( x) dx  lim  0 a c f (x)dx  b Example 6 a) Show that 2 1 1 x dx is improper. b) Find the Cauchy’s principle value of 2 1
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