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real integrals - Chapter 7 Evaluation of Improper Integrals...

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Chapter 7: Evaluation of Improper Integrals Advice 1: Page No. 257: Q. Nos.: 1 - 5
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exists. RHS on limit the provided ) ( lim ) ( then 0, x all for continuous is f(x) Let ) 1 ( 0 0 = R R dx x f dx x f
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( 29 - ) ( then x. all for continuous is f(x) Let 2 dx x f exist. RHS on limits both the provided , ) ( lim ) ( lim 0 1 2 0 2 1 - + = R R R R dx x f dx x f
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exist. RHS. on limit the provided , ) ( lim dx f(x) P.V. number the is (2) integral the of (P.V.) value principal Cauchy R R - - R = dx x f
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dx x f dx x f ) ( P.V. of existence the implies ) ( integral improper of Existence (1) : Remark - not true. is converse But
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0 2 x lim xdx lim dx ) x ( f P.V. x.Then f(x) Let . Ex R R 2 R R R - R = = = = - -
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2 lim 2 lim lim lim ) ( 2 2 2 2 1 1 2 0 2 0 1 1 R R xdx xdx dx x f But R R R R R R - - + - = + =
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- . exist to fails dx ) x ( f integral improper The exist to fails RHS on Limit
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If the function f(x) ( ) is an even function i.e. f(-x) =f(x) for all x, then the symmetry of the graph of y = f(x) with respect to y axis leads to < < - x = - R R R dx x f dx x f 0 ) ( 2 ) (
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When f(x) is an even function and the Cauchy pricncipal value exists, then - - = = 0 ) ( 2 ) ( ) ( . . dx x f dx x f dx x f V P
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To evaluate improper integral of Even Rational Functions f(x)=p(x)/ q(x) p(x) and q(x) are polynomials with real coefficients and no factors in common q(z) has no real zeros but has at least one zero above the real axis.
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Method Identify all distinct zeros of the polynomial q(z) that lie above the real axis They will be finite in number May be labeled as z 1 , z 2 ,…..z n where n is less than or equal to the degree of q(z) Now, integrate the quotient f(z)=p(z)/q(z) around the positively oriented boundary of the semicircular region.
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The simple closed contour consists of The segment of the real axis from z = -R to z =R and The top half of the circle R z = described counterclockwise and denoted by C R .
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Remark: The positive number R is large enough that the points z 1 , z 2 ,… z n all lie inside closed path.
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.
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