Integral (極難之 2, 有興趣挑戰請進)

我知道有人用integrator計到呢個答案出嚟：



圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/unsuggested_answer.jpg



但諗深一層，呢個絕對唔係一個好嘅答案！
其實用人手去仲計簡單得多啦！

　∫√(x3 + 1) dx

= ∫(1 + x3)1/2 dx

= ∫[1 + (1/2)x3 + [((1/2)(- 1/2))/2!](x3)2 + [((1/2)

(- 1/2)(- 3/2))/3!](x3)3 + ……] dx

= ∫[1 + (1/2)x3 - (1/(2．4))x6 + ((1．3)/(2．4．6))

x9 - …… + (- 1)n - 1 [(1．3．5 …… (2n - 3))/(2．4．

6 …… (2n))]x3n + ……] dx

= ∫[1 + (1!!/2!!)x3 - (3!!/4!!)(x6/3) + (5!!/6!!)(x9/5)

- …… + (- 1)n - 1 ((2n - 1)!!/(2n)!!)(x3n/(2n - 1))

+ ……] dx

= ∫[1 + Σ(n = 1 to ∞)[((- 1)n - 1 (2n - 1)!! x3n)/

((2n - 1)(2n)!!)]] dx

= x + Σ(n = 1 to ∞)[((- 1)n - 1 (2n - 1)!! x3n + 1)/

((3n + 1)(2n - 1)(2n)!!)] + C

雖然我f.3都識f.6的calculas,
但係真係估唔到個答案會咁煩= =
我以為用u-sub就得架啦 ==

let u = 開方x+1 = y
仲有你係[子文]?

2007-10-15 17:09:49 補充：
yes or no .................???

This is an elliptic integral.

We can check here:
http://integrals.wolfram.com

Elliptic integral:
http://en.wikipedia.org/wiki/Elliptic_integral