CALCULUSIICH13

# 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.
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
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
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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