Re: [積分] 中興的題目
※ 引述《damaka (九死一生之路)》之銘言:
: 1. 求 ∫ x^5 e^(-x) dx
: 2. tan^-1 (99) (0)
: 題目是對arctan (0) 的99階 微分
: 謝謝
1. ∫(x^5)*(e^(-x)) dx
= (-e^(-x))*(x^5) - ∫(-e^(-x))*(5*x^4)dx
= (-x^5)*(e^(-x)) + 5*∫(x^4)*(e^(-x)) dx
= (-x^5)*(e^(-x)) + 5*[(-e^(-x))*(x^4) - ∫(-e^(-x))*(4*x^3) dx]
= (-x^5 - 5*x^4)*(e^(-x)) + 20*∫(x^3)*(e^(-x)) dx
= (-x^5 - 5*x^4)*(e^(-x)) + 20*[(-e^(-x))*(x^3) - ∫(-e^(-x))*(3*x^2) dx]
= -(x^5 + 5*x^4 + 20*x^3)*(e^(-x)) + 60*∫(x^2)*(e^(-x)) dx
= -(x^5 + 5*x^4 + 20*x^3)*(e^(-x)) + 60*[(-e^(-x))*(x^2) - ∫(-e^x)*(2x) dx]
= -(x^5 + 5*x^4 + 20*x^3 + 60*x^2)*(e^(-x)) + 120*∫x*(e^(-x)) dx
= -(x^5 + 5*x^4 + 20*x^3 + 60*x^2)*(e^(-x))
+ 120*[(-e^(-x))*x - ∫(-e^(-x)) dx]
= -(x^5 + 5*x^4 + 20*x^3 + 60*x^2 + 120x + 120)*e^(-x) + c
-1 x 1
2. tan x = ∫ ------- dx
0 1 + t^2
x
= ∫ (1 - t^2 + t^4 - t^6 + ... + ((-1)^n]*t^(2n)) + ...) dt
0
t^3 t^5 t^7 ((-1)^n)*(t^(2n+1)) |x
= (t - --- + --- - --- + ... + ------------------- |
3 5 7 2n + 1 |0
x^3 x^5 x^7 ((-1)^n)*(x^(2n+1))
= x - --- + --- - --- + ... + ------------------- + ... |x|<1
3 5 7 2n + 1
1 1 1 (-1)^n
x = 1 時, 1 - --- + --- - --- + ... + ------ + ... 收斂
3 5 7 2n + 1
-1 π ∞ (-1)^n π
且收歛於 tan 1 = --- , 即 Σ ------ = ---
4 n=0 2n + 1 4
1 1 1 (-1)^(n+1)
x = -1 時, -1 + --- - --- + --- + ... + ---------- + ... 收斂
3 5 7 2n + 1
-1 π
且收斂於 tan (-1) = - ---
4
-1 ∞ ((-1)^n)*(x^(2n+1))
所以 f(x) = tan x = Σ ------------------- , |x|≦1
n=0 2n + 1
(2n+1)
f (0) (-1)^n (2n+1)
------------ = ------- 所以 f (0) = ((-1)^n)*(2n)!
(2n + 1)! 2n + 1
(99)
2n + 1 = 99 , n = 49 所以 f (0) = ((-1)^49)*(98)! = -98!
-1
所以 tan (0) 的99階微分為 -98!
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