作者LuisSantos (^______^)
看板trans_math
標題Re: [積分]請問一些部份積分
時間Fri Oct 5 19:27:24 2007
※ 引述《kissmeeggy (eggy)》之銘言：
: 7-x-2x^2
: 一: ∫---------------- dx
: (x-1)^2(x^2+x+2 )
: x^3+4x^2-4x-1
: 二: ∫-----------------dx
: (x^2+1)^2
: x
: 三: ∫---------------dx
: (x+1)^2(x^2+1)
x A B Cx + D
令 ------------------ = ----- + --------- + ---------
((x+1)^2)(x^2 + 1) x + 1 (x + 1)^2 x^2 + 1
x = (A)(x + 1)(x^2 + 1) + (B)(x^2 + 1) + (Cx + D)((x + 1)^2)
= (A + C)(x^3) + (A + B + 2C + D)(x^2) + (A + C + 2D)(x) + (A + B + D)
A + C = 0 => C = -A ------(1)
A + B + 2C + D = 0 ------(2)
A + C + 2D = 1 ------(3)
A + B + D = 0 ------(4)
(1)代入(2)、(3)得 A + B - 2A + D = 0 => -A + B + D = 0 ------(5)
1
A - A + 2D = 1 => 2D = 1 => D = --- 代入(4)、(5)得
2
1 -1
A + B + --- = 0 => A + B = --- ------(6)
2 2
1 -1
-A + B + --- = 0 => -A + B = --- ------(7)
2 2
(6) - (7) => 2A = 0 => A = 0 代入(1)得 C = -A = 0
-1
(6) + (7) => 2B = -1 => B = ---
2
x -1 1 1 1
-------------------- = (---)(---------) + (---)(-------)
((x + 1)^2)(x^2 + 1) 2 (x + 1)^2 2 x^2 + 1
x
∫-------------------- dx
((x + 1)^2)(x^2 + 1)
-1 1 1 1
= ∫(---)(---------) + (---)(-------) dx
2 (x + 1)^2 2 x^2 + 1
1 1 1 -1
= (---)(-----) + (---)(tan (x)) + c
2 x + 1 2
: x^2+2x-1
: 四: ∫-------------dx
: 2x^3+3x^2-2x
--
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