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CALCULUSIICH15

# 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.
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.
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
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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