M408C - Notes - Integrals for Final

M408C - Notes - Integrals for Final - Contents 2 Exam...

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Unformatted text preview: Contents 2 Exam Review: Long Division and Complete-the-Square Example 3 Finishing Off the Complete-the-Square Example 4 The Integration by Parts example 5 Which also involves trig sub 6 Trig Integrals 7 A Third Trig Integral 8 Trig Sub 9 The Last Trig Subbie Exam Review: Long Division and Complete-the-Square Example Here it is, the exam review sheet. The topics to be covered are: a) Long division; completing the square b) Integration by parts c) Trig Integrals d) Trig substitution e) Partial fractions A) Example R x 3 +1 x 2 +2 x +3 dx . When you have a quotient of polynomials, it’ll be either a complete the square problem or a partial fractions, but in either case you’ll need to do a long division first. Then Z x 3 + 1 x 2 + 2 x + 3 dx = Z ( x- 2) + x + 7 x 2 + 2 x + 3 ¶ dx What happens next depends whether the denominator factors or not: wherether b 2- 4 ac ≥ 0 or b 2- 4 ac < 0. But a = 1 ,b = 2 ,c = 3, and b 2- 4 ac = 2 2- 4(1)(3) < 0 so the denominator does not factor, and this is a complete-the-squares problem. Completing the square: x 2 +2 x +3 = ( x + e ) 2 + f = x 2 +2 ex +( e 2 + f ), so ya got 2 e = 2 and e 2 + f = 3. Then e = 1 and 1+ f = 3 or f = 2. In all, x 2 + 2 x + 3 = ( x + 1) 2 + 2 and Z x 3 + 1 x 2 + 2 x + 3 dx = Z ( x- 2) + x + 7 x 2 + 2 x + 3 ¶ dx = Z ( x- 2) dx + Z x + 7 ( x + 1) 2 + 2 dx = x 2 2- 2 x ¶ + Z x...
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This note was uploaded on 03/22/2008 for the course M 408c taught by Professor Mcadam during the Fall '06 term at University of Texas.

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M408C - Notes - Integrals for Final - Contents 2 Exam...

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