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CALCULUSIICH15 - C H APT ER 1 F U RT H ER T ECH N I QU ES I...

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CHAPTER 1 FURTHER TECHNI QUES I N CONSTRUCTI NG ANTI DERI VATI VES
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CONTENTS 1.1. Review of Definite and Indefinite I ntegrals 1.2. Trigonometric Integrals 1.3. Partial Fractions 1.4. I ntegration Using Tables and Computer Algebra Systems 1.5. I mproper I ntegrals.
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1.5. IMPROPER I NTEGRALS 1.5.1. Type 1: Infinite Intervals 1.5.2. Type 2: Discontinuous Integrands 1.5.3. Comparison Test for Improper I ntegrals
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1.5.1 Type 1: I nfinite I ntervals Consider the area A of the infinite region S under the curve y =1/ x 2 , above the x - axis and on the right of x =1. At a first look, it seems that A is infinite.
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However if we look at the part of S on the left of the line x = t . then 1 1 1 1 1 ) ( 1 1 2 < - = - = = t x dx x t A t t 1 1 1 lim ) ( lim = - = t t A t t Moreover So we say that the area of the infinite region S is 1, and 1 1 lim 1 1 2 1 2 = = t t dx x dx x
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Definition of Improper Integral of Type 1. (a) If exists for all t a , then provided the limit exists (as a finite number) (b) If exists for all t b , then t a dx x f ) ( = t a t a dx x f dx x f ) ( lim ) ( b t dx x f ) ( -∞ - = b t t b dx x f dx x f ) ( lim ) ( a dx x f ) ( - b dx x f ) ( We say that the improper integrals and are convergent . Otherwise we say that they are divergent
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Definition of Improper Integral of Type 1. (c) If both and are convergent, then we define the value on the right hand side does not depend on the choice of a - - + = a a dx x f dx x f dx x f ) ( ) ( ) ( a dx x f ) ( - a dx x f ) ( These improper integrals may be interpreted as areas of infinite regions if f ( x ) 0
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Example.
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