Partial Fractions Integration: Solving (x^2 - 9)^2 dx, Assignments of Mathematics

Download Partial Fractions Integration: Solving (x^2 - 9)^2 dx and more Assignments Mathematics in PDF only on Docsity! MATH 3B: HOMEWORK 4, SECTION 7.3, #49 Now that I have remembered how to multiply and do partial fractions, lets look at problem 49. We want to compute ∫ 1 (x2−9)2 dx, so let us use the partial fraction decomposition. Since (x 2 − 9)2 = (x + 3)2(x − 3)2, we may write 1 (x + 3)2(x − 3)2 = Ax + B (x + 3)2 + Cx + D (x − 3)2 . Rewriting the right hand side over a single denominator gives 1 (x + 3)2(x − 3)2 = (Ax + B)(x − 3)2 + (Cx + D)(x + 3)2 (x + 3)2(x − 3)2 . We expand and equate the numerators to get 1 = Ax3 + (B − 6A)x2 + (9A − 6B)x + 9B + Cx3(6C + D)x2(9C + 6D)x + 9D. We equate the coefficients of the different powers of x on the left and right side of this equation to get A + C = 0(1) (B − 6A) + (6C + D) = 0(2) (9A − 6B) + (9C + 6D) = 0(3) 9B + 9D = 1.(4) The first and last equation look the simplest, so let use rewrite them as C = −A and D = (1−9B)/9. We then plug these into equations (2) and (3) to get B − 6A + 6(−A) + 1 − 9B 9 = 0(5) 9A − 6B + 9(−A) + 6 ( 1 − 9B 9 ) = 0.(6) Once we collect like terms in equations (5) and (6) (and multiply by 9 to get remove the fraction) we get 1 − 108A = 0(7) 18B − 1 = 0.(8) Conveniently, these equations pretty much solve themselves, since A disappears in (8) and B disappears in (7). Looking back at equations (1) and (4), combined with (7) and (8) we see that A = 1 108 B = 1 18 C = −1 108 D = 1 18 . Now that we have these values, we see that 1