CALCULUSIICH13 - CHAPTER 1 FURTHER TECHNI QUES I N...

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CHAPTER 1 FURTHER TECHNIQUES IN CONSTRUCTING ANTIDERIVATIVES
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CONTENTS 1.1. Review of Definite and Indefinite Integrals 1.2. Trigonometric Integrals 1.3. Partial Fractions 1.4. Integration Using Tables and Computer Algebra Systems 1.5. Improper Integrals.
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1.3. PARTIAL FRACTIONS 1.3.1. Partial Fractions 1.3.2. Case I: simple Poles 1.3.3. Case II: Multiple Poles 1.3.4. General Case 1.3.5. Rationalizing Subsitutions
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1.3.1 Partial Fractions The rational functions are expressed as sum of partial fractions before being integrated. Following are some examples of decomposition into partial fractions 4 1 1 4 4 2 . 3 1 9 ) 2 ( 3 2 ) 2 ( 9 1 ) 1 ( ) 2 ( . 2 1 2 1 1 3 1 2 4 5 . 1 2 3 2 2 2 2 + - + = + + - - + + + + - = - + - - + = - + - x x x x x x x x x x x x x x x x x x
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The partial fractions in (1) and (2) are easily to be integrated since they are of the form Ν - m a x c m , ) ( and 1 if ) )( 1 ( ) ( , 1 if ) ln( ) ( 1 + - - - = - = + - = - - m C a x m c dx a x c and m C a x c dx a x c m m m
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To integrated the second partial fraction in 3. we write 4 1 4 4 1 2 2 2 + - + = + - x x x x x Now to integrate the first partial fraction we make the substitution u = x 2 + 4, du = 2 xdx C x du u dx x x + + = = + ) 4 ln( 1 4 2 2 1 2 1 2 Now to integrate the second partial fraction we make the substitution x = 2 u , dx = 2 du C x u du dx x + = + = + - ) 2 ( tan 1 4 1 1 2 1 2 2 1 2
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In general. Let f be a rational function ) ( ) ( ) ( x Q x P x f = Assume that P and Q have no common factors f is said to be proper if deg( P ) < deg( Q ). In case f is not proper, we can carry out long division for P ( x ) by Q ( x ) to obtain the quotient S ( x ) and the remainder R ( x ) ) ( ) ( ) ( ) ( ) ( ) ( x Q x R x S x Q x P x f + = = Now S ( x ) is a polynomial that is easily integrated. On the other hand R ( x )/ Q ( x ) is a proper rational
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Finally these partial fractions are integrated by making suitable substitution Now to decompose a proper rational function f as a sum of partial fractions, we first decompose the denominator as a product of irreducible factors,
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CALCULUSIICH13 - CHAPTER 1 FURTHER TECHNI QUES I N...

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