Re: [積分] 積分一題
※ 引述《le5868ov (我一定要上榜！~!~)》之銘言：
: 3
: x + 2x + 1
: ∫ ------------- dx 怎麼求? 麻煩了！
: 4
: (x + 2)
x^3 + 2x + 1 A B C D
令 -------------- = ----- + --------- + --------- + ---------
(x + 2)^4 x + 2 (x + 2)^2 (x + 2)^3 (x + 2)^4
則 x^3 + 2x + 1 = (A)((x + 2)^3) + (B)((x + 2)^2) + (C)(x + 2) + D
= (A)(x^3) + (6A + B)(x^2) + (12A + 4B + C)(x) +
(8A + 4B + 2C + D)
A = 1 ------(1)
6A + B = 0 ------(2)
12A + 4B + C = 2 ------(3)
8A + 4B + 2C + D = 1 ------(4)
A = 1 代入(2)得 6 + B = 0 => B = -6
A = 1 , B = -6 代入(3)得 12 - 24 + C = 2 => C = 14
A = 1 , B = -6 , C = 14 代入(4)得
8 - 24 + 28 + D = 1 => D = -11
x^3 + 2x + 1 1 -6 14 -11
-------------- = ----- + --------- + --------- + ---------
(x + 2)^4 x + 2 (x + 2)^2 (x + 2)^3 (x + 2)^4
x^3 + 2x + 1
∫------------ dx
(x + 2)^4
1 -6 14 -11
= ∫----- + --------- + --------- + --------- dx
x + 2 (x + 2)^2 (x + 2)^3 (x + 2)^4
6 7 11 1
= ln|x + 2| + ----- - --------- + (----)(---------) + c
x + 2 (x + 2)^2 3 (x + 2)^3
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06/11 13:15, , 1F
06/11 13:15, 1F
※ 編輯: LuisSantos 來自: 140.119.66.29 (06/11 13:41)
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