Re: [積分]
※ 引述《gwish (曖曖內含光)》之銘言：
: 2x+2
: ∫--------------dx
: (x-1)^3(x^2+1)
: 這題部分分式一直算錯
: 不知道是不是哪裡搞錯了
: 謝謝
2x + 2 A B C Dx + E
令 -------------------- = ----- + ------- + ------- + --------
((x-1)^3)(x^2 + 1) x - 1 (x-1)^2 (x-1)^3 x^2 + 1
則 2x + 2 = (A)((x-1)^2)(x^2 + 1) + (B)(x-1)(x^2 + 1) + (C)(x^2 + 1)
+ (Dx + E)((x-1)^3)
= (A)(x^2 - 2x + 1)(x^2 + 1)
+ (B)(x-1)(x^2 + 1) + (C)(x^2 + 1)
+ (Dx + E)(x^3 - (3)(x^2) + 3x - 1)
= (A)(x^4 - (2)(x^3) + (2)(x^2) - 2x + 1)
+ (B)(x^3 - x^2 + x - 1) + (C)(x^2 + 1) + (D)(x^4) + (-3D+E)(x^3)
+ (3D-3E)(x^2) + (-D+3E)(x) - E
= (A + D)(x^4) + (-2A + B - 3D + E)(x^3)
+ (2A - B + C + 3D - 3E)(x^2)
+ (-2A + B - D + 3E)(x) + (A - B + C - E)
= 2x - 2
A + D = 0 ------(1)
-2A + B - 3D + E = 0 ------(2)
2A - B + C + 3D - 3E = 0 ------(3)
-2A + B - D + 3E = 2 ------(4)
A - B + C - E = 2 ------(5)
(1) => D = -A ------(6)
(6)代入(2)、(3)、(4)得
A + B + E = 0 ------(7)
-A - B + C - 3E = 0 ------(8)
-A + B + 3E = 2 ------(9)
(7) => E = -A - B ------(10)
(10) 代入 (5)、(8)、(9)得
A - B + C + A + B = 2 => 2A + C = 2 ------(11)
-A - B + C + 3A + 3B = 0 => 2A + 2B + C = 0 ------(12)
-A + B - 3A - 3B = 2 => 4A + 2B = -2 => 2A + B = -1 ------(13)
(11) => C = 2 - 2A 代入(12)
2A + 2B + 2 - 2A = 0 => 2B + 2 = 0 => B + 1 = 0 => B = -1 代入 (13)
2A - 1 = -1 => 2A = 0 => A = 0 代入(11)得 C = 2
A = 0 代入(6)得 D = -A = 0
A = 0 , B = -1 代入(10)得 E = -A - B = 0 - (-1) = 0 + 1 = 1
2x + 2 -1 2 1
------------------ = ------- + ------- + -------
((x-1)^3)(x^2 + 1) (x-1)^2 (x-1)^3 x^2 + 1
2x + 2
∫------------------ dx
((x-1)^3)(x^2 + 1)
-1 2 1
= ∫------- + ------- + ------- dx
(x-1)^2 (x-1)^3 x^2 + 1
1 -1 1 -1
= (-1)(-1)(-----) + (2)(---)(-------) + tan x + c
x - 1 2 (x-1)^2
1 1 -1
= ----- - ------- + tan x + c
x - 1 (x-1)^2
--
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03/28 00:08, , 1F
03/28 00:08, 1F
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