推 XYX16888 :發送100P 04/06 23:26
※ 引述《XYX16888 (XYX)》之銘言：
: 1.Suppose f is continuous on [0,∞) and lim_{x->∞}f(x)=1.
: Is it possible that
: ∞
: ∫ f(x) dx is convergent?
: 0
No. Pick 0.5>e>0. Choose M>0 large such that f(x)>1-e>0.5 for all x>M.
But then for each K>0 and each T>0, we have
∞
∫ f(x)dx > K
T
So the integral does not converge.
: 2.Show that if a>-1 and b>a+1 , then the following
: integral is convergent.
: ∞
: ∫ [ (x^a) / (1 + x^b) ] dx
: 0
: 希望有詳解 謝謝
: 每題100P先搶先贏囉！
x^a / (1+x^b) =< x^a / x^b
Consider
∞ ∞
∫ [ (x^a) / (1 + x^b) ] dx ≦ ∫ x^a/x^b dx which is convergent since b>a+1
1 1
On the other hand,
1 1
∫ [ (x^a) / (1 + x^b) ] dx ≦ ∫ x^a dx which is convergent since a>-1.
0 0
By comparison test, the integral converges
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 219.71.210.134