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barcode in ssrs 2008 and the problem reduces to nding the antiderivative of a proper rational function. in VS .NET
and the problem reduces to nding the antiderivative of a proper rational function. QR Code ISO/IEC18004 Reader In .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Denso QR Bar Code Maker In .NET Using Barcode encoder for .NET framework Control to generate, create Denso QR Bar Code image in .NET framework applications. The theorems that follow hold for polynomials with arbitrary real coef cients. However, for simplicity we shall illustrate them only with polynomials whose coef cients are integers. QR Code Recognizer In Visual Studio .NET Using Barcode decoder for .NET Control to read, scan read, scan image in VS .NET applications. Generating Bar Code In Visual Studio .NET Using Barcode creation for .NET Control to generate, create barcode image in .NET framework applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
Recognize Barcode In .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications. QR Code ISO/IEC18004 Creation In C#.NET Using Barcode creator for .NET Control to generate, create QR Code image in VS .NET applications. THE METHOD OF PARTIAL FRACTIONS
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Linear Barcode Creation In .NET Using Barcode creator for .NET Control to generate, create Linear image in .NET framework applications. ITF Drawer In .NET Using Barcode generator for .NET framework Control to generate, create Interleaved 2 of 5 image in .NET applications. Fig. 401 Theorem 40.1: Any polynomial D(x) with leading coef cient 1 can be expressed as the product of linear factors, of the form x a, and irreducible quadratic factors (that cannot be factored further), of the form x 2 + bx + c, repetition of factors being allowed. As explained in Section 7.4, the real roots of D(x) determine its linear factors. EXAMPLES Draw Code 39 Full ASCII In None Using Barcode creation for Online Control to generate, create Code 3/9 image in Online applications. Creating Barcode In None Using Barcode creator for Online Control to generate, create bar code image in Online applications. (a) (b) x 2 1 = (x 1)(x + 1) Here, the polynomial has two real roots ( 1) and, therefore, is a product of two linear factors. x 3 + 2x 2 8x 21 = (x 3)(x 2 + 5x + 7) The root x = 3, which generates the linear factor x 3, was found by testing the divisors of 21. Division of D(x) by x 3 yielded the polynomial x 2 + 5x + 7. This polynomial is irreducible, since, by the quadratic formula, its roots are x= which are not real numbers. b 5 3 b2 4c = 2 2 Matrix 2D Barcode Maker In Java Using Barcode maker for Java Control to generate, create 2D Barcode image in Java applications. USS Code 39 Decoder In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Theorem 40.2 (Partial Fractions Representation): Any (proper) rational function f (x) = N(x)/D(x) may be written as a sum of simpler, proper rational functions. Each summand has as denominator one of the linear or quadratic factors of D(x), raised to some power. By Theorem 40.2, f (x)dx is given as a sum of simpler antiderivatives antiderivatives which, in fact, can be found by the techniques already known to us. It will now be shown how to construct the partial fractions representation and to integrate it term by term. Case 1: D(x) is a product of nonrepeated linear factors. The partial fractions representation of f (x) is N(x) A1 A2 An + + + = (x a1 )(x a2 ) (x an ) x a1 x a2 x an The constant numerators A1 , . . . , An are evaluated as in the following example. EXAMPLE EAN13 Drawer In VS .NET Using Barcode creation for ASP.NET Control to generate, create UPC  13 image in ASP.NET applications. Encode Code 128 In Java Using Barcode printer for Android Control to generate, create Code 128A image in Android applications. A1 A2 2x + 1 = + (x + 1)(x 1) x+1 x 1 A A 2x + 1 = (x + 1)(x 1) 1 + (x + 1)(x 1) 2 (x + 1)(x 1) x+1 x 1 2x + 1 = A1 (x 1) + A2 (x + 1) In (1), substitute individually the roots of D(x). With x = 1, 1 = A1 ( 2) + 0 or A1 = 1 2 (1) Printing UCC  12 In Java Using Barcode creator for BIRT reports Control to generate, create UCC  12 image in BIRT applications. Decoding Bar Code In Java Using Barcode Control SDK for BIRT Control to generate, create, read, scan barcode image in Eclipse BIRT applications. Clear the denominators by multiplying both sides by (x + 1)(x 1), (x + 1)(x 1) CHAP. 40] THE METHOD OF PARTIAL FRACTIONS
and with x = 1, 3 2 With all constants known, the antiderivative of f (x) will be the sum of terms of the form 3 = 0 + A2 (2) or A2 = A dx = A ln x a x a Case 2: D(x) is a product of linear factors, at least one of which is repeated. This is treated in the same manner as in Case 1, except that a repeated factor (x a)k gives rise to a sum of the form A2 Ak A1 + + + 2 x a (x a) (x a)k EXAMPLE Multiply by (x l)2 (x 2), 3x + 1 = A1 (x 1)(x 2) + A2 (x 2) + A3 (x 1)2 Letting x = 1, 4 = 0 + A2 ( 1) + 0 Letting x = 2, 7 = 0 + 0 + A3 (1) or A3 = 7 The remaining numerator, A1 , is determined by the condition that the coef cient of x 2 on the right side of (2) be zero (since it is zero on the left side). Thus, A 1 + A3 = 0 or A1 = A3 = 7 or A2 = 4 (2) A3 3x + 1 A1 A2 + + = x 1 (x 1)2 x 2 (x 1)2 (x 2) [More generally, we use all the roots of D(x) to determine some of the A s, and then compare coef cients of as many powers of x as necessary to nd the remaining A s.] Now the antiderivatives of f (x) will consist of terms of the form ln x a plus at least one term of the form A B dx = j (x a) (x a) j 1 Case 3: D(x) has irreducible quadratic factors, but none is repeated. In this case, each quadratic factor x 2 + bx + c contributes a term x2 to the partial fractions representation. EXAMPLE

