q2solns - Mathematics 104 Quiz 2, Friday, Feb 22, 2008 1....

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Unformatted text preview: Mathematics 104 Quiz 2, Friday, Feb 22, 2008 1. Find dx . (x + 2)(x2 + 1) A Bx + C 1 = + 2 Ax2 + A + Bx2 + 2Bx + Cx + 2C = 1 + 0x + 0x2 . 2 + 1) (x + 2)(x x+2 x +1 This gives the system A + B = 0, 2B + C = 0 and A + 2C = 1 which we solve to find A = 1/5, B = -1/5 and C = 2/5. So our integral becomes 1/5 1 dx + x+2 5 2-x 1 1 2 = ln |x + 2| - ln(x2 + 1) + arctan x + C 2+1 x 5 10 5 x2 dx. 2. Find x2 - x - 12 The rational function we need to integrate is not proper. We can rewrite the integrand as x2 x2 - x - 12 x + 12 x + 12 = 2 + 2 =1+ x2 - x - 12 x - x - 12 x - x - 12 (x - 4)(x + 3) So our integral becomes 1 dx + where A(x + 3) + B(x - 4) = x + 12 Substituting x = -3 and x = 4 gives A = 16/7 and B = -9/7. So our answer is x+ 3. Find 16 9 ln |x - 4| - ln |x + 3| + C 7 7 A dx + x-4 B dx x+3 x dx. + 2x + 10 The quadratic polynomial in the denominator is irreducible. So we complete the square: x2 x2 + 2x + 10 = x2 + 2x + 1 + 9 = (x + 1)2 + 9. Then we make the substitution u = x + 1 which gives x2 x dx = + 2x + 10 u-1 du = u2 + 9 u2 u du - +9 u2 1 +9 x+1 3 Thus the integral is 1 1 1 1 ln(u2 + 9) - arctan(u/3) + C = ln(x2 + 2x + 10) - arctan 2 3 2 3 +C ...
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This note was uploaded on 04/07/2008 for the course MAT 104 taught by Professor Edwardnelson during the Spring '08 term at Princeton.

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