作者LuisSantos (^______^)
看板trans_math
標題Re: [積分] 不定積分
時間Sun Dec 3 16:09:26 2006
※ 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
--
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◆ From: 61.66.173.21
1F：推 GayerDior:謝謝... 61.229.152.5 12/03 17:00
2F：推 mathqaz:哇~還真冗長阿。220.138.115.140 12/03 20:23