求解这道不定积分题 急!1.∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx = ?2 ∫ {(x^4 - 3x^2)/ [(x+5)^2] * (x^2-3x+3)^2} dx = ? 最好能把过程写出来 先谢谢好心人了

求解这道不定积分题 急!1.∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx = ?2 ∫ {(x^4 - 3x^2)/ [(x+5)^2] * (x^2-3x+3)^2} dx = ? 最好能把过程写出来 先谢谢好心人了
求解这道不定积分题 急!
1.∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx = ?
2 ∫ {(x^4 - 3x^2)/ [(x+5)^2] * (x^2-3x+3)^2} dx = ?
最好能把过程写出来 先谢谢好心人了

求解这道不定积分题 急!1.∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx = ?2 ∫ {(x^4 - 3x^2)/ [(x+5)^2] * (x^2-3x+3)^2} dx = ? 最好能把过程写出来 先谢谢好心人了
1,
∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx =
= ∫{x^5-3x^4 - 3x^3 + 9x^2)/ [(x+5)^3]}dx
设g(x) = x^5 - 3x^4 - 3x^3 + 9x^2 = (x+5)^5 + a(x+5)^4 + b(x+5)^3 + c(x+5)^2 + e(x+5) + f,
f=g(-5)=(-5)^5 - 3*5^4 - 3(-5)^3 + 9*5^2 = -8*5^4 + 3*5^3 + 9*5^2
= -37*5^3 + 9*5^2 = (9-5*37)25= -4400.
g'(x) = 5x^4 - 12x^3 - 9x^2 + 18x = 5(x+5)^4 + 4a(x+5)^3 + 3b(x+5)^2 + 2c(x+5) + e
e = g'(-5) = 5*5^4 - 12(-5)^3 - 9*5^2 + 18(-5) = 37*5^3 - 9*5^2 - 90
= -4400-90 = -4490
g''(x) = 20x^3 - 36x^2 - 18x + 18 = 20(x+5)^3 + 12a(x+5)^2 + 6b(x+5) + 2c,
c = g''(-5)/2 = 10(-5)^3 - 18*5^2 - 9(-5)+ 9 = -68*25 + 50 = -13550
g'''(x) = 60x^2 - 72x - 18 = 60(x+5)^2 + 24a(x+5) + 6b,
b = g'''(-5)/6 = 10*5^2 - 12(-5) - 3 = 250+60-3=307
g^(4)(x) = 120x - 72 = 120(x+5) + 24a,
a = g^(4)(-5)/24 = 5(-5)-3=-28
∫ {(x^4 - 3x^2)/ [(x+5)^3] * (x-3)} dx =
= ∫{x^5-3x^4 - 3x^3 + 9x^2)/ [(x+5)^3]}dx
= ∫{(x+5)^2-28(x+5)+307-13550/(x+5)-4490/(x+5)^2-4400/(x+5)^3}dx
= (x+5)^3/3 -14(x+5)^2 + 307(x+5)-13550ln|x+5|+4490/(x+5)+2200/(x+5)^2 + C
C = const.
2,
也按照题1的方法做,用待定系数法.
要不,你自个练练?

(1)=1/3*x^3-9*x^2+192*x-1646*log(x+5)-4310/(x+5)+2200/(x+5)^2
= 3 2 4310 2200
1/3 x - 9 x + 192 x - 1646 log(x + 5) - ----- + --------<...

全部展开

(1)=1/3*x^3-9*x^2+192*x-1646*log(x+5)-4310/(x+5)+2200/(x+5)^2
= 3 2 4310 2200
1/3 x - 9 x + 192 x - 1646 log(x + 5) - ----- + --------
x + 5 2
(x + 5)
(2)=1/7*x^7-8/3*x^6+147/5*x^5-535/2*x^4+6989/3*x^3-21543*x^2+256108*x-1483930*log(x+5)-1016950/(x+5)
= 7 6 5 4 3 2
1/7 x - 8/3 x + 147/5 x - 535/2 x + 6989/3 x - 21543 x + 256108 x
1016950
- 1483930 log(x + 5) - -------
x + 5
第一行为答案，第二行为美化后的答案

收起