興趣 Integral 之 2

It can be shown that if
x²
F(x)＝∫√(t^4＋x³)dt (t^4 instead of t² in the first integral)
0
then
x²
dF(x)/dx＝√(t^8＋x³)dt＋∫{3x²/[2√(t^4＋x³)]}dt (the second)
0

2008-06-26 03:44:17 補充：
It can be shown that if
　　　x²
F(x)＝∫√(t^4＋x³)dt　　　(t^4 instead of t² in the first integral)
　　　0
then
　　　　　　　　　　　x²
dF(x)/dx＝2x√(t^8＋x³)＋∫{3x²/[2√(t^4＋x³)]}dt　　(exactly the second integral)
　　　　　　　　　　　0

頭先打錯。

2008-06-26 04:08:58 補充：
It can be shown that if
　　　x²
F(x)＝∫√(t^4＋x³)dt　　　(t^4 instead of t² in the first integral)
　　　0
then
　　　　　　　　　　　x²
dF(x)/dx＝2x√(x^8＋x³)＋∫{3x²/[2√(t^4＋x³)]}dt　　(exactly the second integral)
　　　　　　　　　　　0

又打錯。

2008-06-26 22:24:04 補充：
　　　　 x²
Let F(x)＝ ∫ √(t^4＋x³)dt　　　(t^4 instead of t²)
　　　　 0

　　f(x,t)＝√(t^4＋x³)
　　a(x)＝0
　　b(x)＝x²

By Leibniz integral rule:

　　 b(x)　　　　　　　　　　　　　　　　　　　 b(x)
(d/dx) ∫ f(x,t)dt＝f[x,b(x)](∂/∂x)b(x)－f[x,a(x)](∂/∂x)a(x)＋ ∫ (∂/∂x)f(x,t)dt
　　 a(x)　　　　　　　　　　　　　　　　　　　 a(x)

we have
　　　　　　　　　　　　　　　　　　　 x²
(d/dx)F(x)＝f(x, x²)(∂/∂x)(x²)－f(x,0)(∂/∂x)(0)＋ ∫ (∂/∂x)[√(t^4＋x³)]dt
　　　　　　　　　　　　　　　　　　　 0
　　　　　　　　　　 x²
＝√[(x²)^4＋x³](2x)－0＋ ∫ [1/√(t^4＋x³)](3x²)dt
　　　　　　　　　　 0
　　　　　　　 x²
＝2x√(x^8＋x³)＋ ∫ {3x²/[2√(t^4＋x³)]}dt
　　　　　　　 0

這條問題是由邊個Yahoo!奇摩知識+會員問你嫁？
在Yahoo!奇摩知識+有冇這條問題？

This is not true.

2008-06-24 17:00:26 補充：
嗯，我用機int過，結果係
}} int('sqrt(t^2+x^3)',t)
.
ans =
.
1/2*t*(t^2+x^3)^(1/2)+1/2*x^3*log(t+(t^2+x^3)^(1/2))
.
}} int('1/sqrt(t^4+x^3)',t)
.
ans =
.
1/(i/x^(3/2))^(1/2)*(1-i/x^(3/2)*t^2)^(1/2)*(1+i/x^(3/2)*t^2)^(1/2)/(t^4+x^3)^(1/2)*EllipticF(t*(i/x^(3/2))^(1/2),i)
.
It becomes a monster!

2008-06-26 15:54:27 補充：
differentiation 可以咁樣走入去 integral 入面??
dF(x)/dx＝2x√(t^8＋x³) 呢部份用左 Fundamental theorem http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
但 √(t^4＋x³) 係 f(t,x) 唔係單純的 f(t), 唔....

Yup, dt is missing in the last
A Taiwan Yahoo knowledge fd asked me this Q

2008-06-23 17:16:40 補充：
Yahoo!奇摩知識+曾有此發問，但最終因無人答而移除．
此題也困擾了末將成個禮拜，向來源者求証過這是書的答案．
當然，書亦有可能錯，末將亦承認．但諸位高手是否有了心目中的答案？不妨發表一下
THX

sounds fun~
where did you get this question from?

2008-06-23 05:49:29 補充：
This is not true x 2

2008-06-23 05:54:40 補充：
It is easy to see, from a very very rough estimate:
The degree of the first statement is of order t^2 -> x^4
while the degree of the second statement is of order x^5, so when x gets large, the 2x rt(x^8+x^3) term will become dominant and gets a lot bigger.

2008-06-24 00:05:01 補充：
now it is clear that the statement is wrong, our task is not to try to show it, but instead think of how to correct the statement as little as possible.

2008-06-24 00:05:04 補充：
(try plug in some x in math software will easily see that the statement is wrong, and agree with my observation above that the 2x rt(x^8+x^3) is too large)

2008-06-24 00:09:14 補充：
but ignoring the fact that the statement is wrong, the most direct method is to differentiate both side w.r.t. x (using chain rule) and compare. If the statement is correct, after differentiating, they will differ only by a constant which is 0 by plug in x=0

2008-06-26 01:21:11 補充：
相信nychan63 的改正是對的~
用 chain rule 會出埋開頭的 2x
希望有人可以寫低個 detail 啦~ 我唔識post 圖...

Is "dt" missed in the last integration?