Re: [積分] 不定積分

看板trans_math作者LuisSantos (^______^)時間19年前 (2006/12/03 15:36)推噓2(2推 0噓 0→)

留言2則, 2人參與討論串8/45 (看更多)

※ 引述《GayerDior ( 四歲就很帥......)》之銘言： : x^3 - 1 : 1. ∫------------ dx : x^2(x-2)^3 : 3x+1 : 2. ∫---------- dx : (x^2-4)^2 : 如果有人會寫， : 請幫我寫出詳細的過程， : 謝謝了... x^3 - 1 A B C D E 1. 令 ---------------- = --- + --- + --- + ------- + ------- (x^2)((x-2)^3) x x^2 x-2 (x-2)^2 (x-2)^3 x^3 - 1 = (A)(x)((x-2)^3) + (B)((x-2)^3) + (C)(x^2)((x-2)^2) + (D)(x^2)(x-2) + (E)(x^2) = (A)(x)(x^3 - 6x^2 + 12x - 8) + (B)(x^3 - 6x^2 + 12x - 8) + (C)(x^2)(x^2 - 4x + 4) + (D)(x^2)(x-2) + (E)(x^2) = (A + C)(x^4) + (-6A + B -4C + D)(x^3) + (12A - 6B + 4C - 2D + E)(x^2) + (-8A + 12B)(x) + (-8B) A + C = 0 ---------(1) -6A + B - 4C + D = 1 ---------(2) 12A - 6B + 4C - 2D + E = 0 ---------(3) -8A + 12B = 0 ---------(4) -8B = -1 ---------(5) 1 (5) => B = --- 代入 (2) , (3) , (4) 8 1 -7 -6A + --- - 4C + D = 1 => 6A + 4C - D = ---- -------(6) 8 8 3 3 12A - --- + 4C - 2D + E = 0 => 12A + 4C - 2D + E = --- -------(7) 4 4 3 3 3 -8A + --- = 0 => 8A = --- => A = ---- 代入(1) , (6) , (7) 2 2 16 3 -3 ---- + C = 0 => C = ---- 16 16 9 -7 9 3 -7 9-6+7 5 --- + 4C - D = ---- => --- - --- - D = ---- => D = ------- = --- 8 8 8 4 8 8 4 9 3 9 3 3 --- + 4C - 2D + E = --- => --- - --- - 2D + E = --- 4 4 4 4 4 9 3 10 3 => --- - --- - ---- + E = --- 4 4 4 4 -9 + 3 + 10 + 3 7 => E = ------------------ = --- 4 4 x^3 - 1 3/16 1/8 -3/16 5/4 7/4 -------------- = ------ + ----- + ----- + ------- + ------- (x^2)((x-2)^3) x x^2 x - 2 (x-2)^2 (x-2)^3 x^3 - 1 ∫-------------- dx (x^2)((x-2)^3) 3/16 1/8 -3/16 5/4 7/4 = ∫(------ + ----- + ----- + ------- + -------) dx x x^2 x - 2 (x-2)^2 (x-2)^3 3 1 1 1 3 1 = (---)(∫--- dx) + (---)(∫--- dx) - (----)(∫----- dx) 16 x 8 x^2 16 x - 2 5 1 7 1 + (---)(∫------- dx) + (---)(∫------- dx) 4 (x-2)^2 4 (x-2)^3 3 1 1 3 5 1 7 1 = (---)(ln|x|) - (---)(---) - (----)(ln|x-2|) + (-)(---) - (---)(-------) + c 16 8 x 16 4 x-2 8 (x-2)^2 -- ※ 發信站: 批踢踢實業坊(ptt.cc) ◆ From: 61.66.173.21

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GayerDior

12/03 17:00, , 1^F

12/03 17:00, 1^F

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mathqaz

12/03 20:23, , 2^F

12/03 20:23, 2^F

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