lec05 - Q ( x ) = ( ax 2 + bx + c ) r = R ( x ) Q ( x ) = A...

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5 - Integration of Rational Functions by Partial Fractions Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 1B Calculus
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Partial Fractions Case 3: Distinct Irreducible Quadratic Factors Q ( x ) = ( ax 2 + bx + c ) · ··· = R ( x ) Q ( x ) = Ax + B ax 2 + bx + c + ··· Example x 2 - 2 x - 1 ( x - 1) 2 ( x 2 + 1) = A x - 1 + B ( x - 1) 2 + Cx + D x 2 + 1 x 2 - 2 x - 1 = A ( x - 1)( x 2 + 1) + B ( x 2 + 1)+ ( Cx + D )( x - 1) 2 = ( A + C ) x 3 + ( - A + B - 2 C + D ) x 2 + ( A + C - 2 D ) x + ( - A + B + D )
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Partial Fractions Example A + C = 0 - A + B - 2 C + D = 1 A + C - 2 D = - 2 - A + B + D = - 1 Solve by elimination = A = 1 ,B = - 1 ,C = - 1 ,D = 1 Z x 2 - 2 x - 1 ( x - 1) 2 ( x 2 + 1) dx = Z 1 x - 1 - 1 ( x - 1) 2 - x - 1 x 2 + 1 dx = ln | x - 1 | + 1 x - 1 - 1 2 ln( x 2 + 1) + tan - 1 x + C
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Integrating Irreducible Quadratic Partial Fractions How to integrate Ax + B ax 2 + bx + c 1) Complete the square and substitute = ax 2 + bx + c = u 2 + d 2) Split numerator into a term with u and a constant term 3) Integration of first term gives ln , second term gives tan - 1 Example Z x + 4 x 2 - 4 x + 6 dx = Z x + 4 ( x - 2) 2 + 2 dx = ± u = x - 2 du = dx ² = Z u + 6 u 2 + 2 du = Z u u 2 + 2 du + Z 6 u 2 + 2 du = 1 2 ln( u 2 + 2) + 6 1 2 tan - 1 u 2 + C = 1 2 ln( x 2 - 4 x + 6) + 3 2 tan - 1 x - 2 2 + C
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Partial Fractions Case 4: Repeated Irreducible Quadratic Factors
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Unformatted text preview: Q ( x ) = ( ax 2 + bx + c ) r = R ( x ) Q ( x ) = A 1 x + B 1 ax 2 + bx + c + A 2 x + B 2 ( ax 2 + bx + c ) 2 + + A r x + B r ( ax 2 + bx + c ) r + Example 2 x + 1 ( x + 1) 3 ( x 2 + 4) 2 = A x + 1 + B ( x + 1) 2 + C ( x + 1) 3 + Dx + E x 2 + 4 + Fx + G ( x 2 + 4) 2 Integrating Squared Quadratic Fractions Z 1 ( x 2 + a 2 ) 2 dx = " x = a tan , sin = x x 2 + a 2 dx = a sec 2 d, cos = a x 2 + a 2 # = Z 1 a 4 sec 4 a sec 2 d = 1 a 3 Z cos 2 d = 1 2 a 3 Z (1 + cos 2 ) d = 2 a 3 + sin 2 4 a 3 + C = 1 2 a 3 tan-1 x a + x x 2 + a 2 a x 2 + a 2 2 a 3 + C = 1 2 a 3 tan-1 x a + x 2 a 2 ( x 2 + a 2 ) + C...
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lec05 - Q ( x ) = ( ax 2 + bx + c ) r = R ( x ) Q ( x ) = A...

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