Re: [微積] 計財103
※ 引述《lyndonxxx (lyndon)》之銘言：
: http://ppt.cc/gnAa
: 今天的考題
: 請問六和七該怎麼想
6. Show that if a > -1 and b > a+1 , then the following integral is convergent
∞ x^a
∫ --------- dx
0 1 + x^b
∞ x^a
proof: ∫ --------- dx
0 1 + x^b
1 x^a ∞ x^a
= ∫ --------- dx + ∫ --------- dx
0 1 + x^b 1 1 + x^b
x^a x^a 1
∵ --------- < ----- = ---------
1 + x^b x^b x^(b-a)
∞ 1
and ∫ --------- dx
1 x^(b-a)
R 1
= lim ∫ --------- dx
R→∞ 1 x^(b-a)
-1 1 |R
= lim (-------)(-----------)| (∵b>a+1 => b-a-1 >0)
R→∞ b-a-1 x^(b-a-1) |1
-1 1 1
= lim (-------)(-----------) + -----------
R→∞ b-a-1 R^(b-a-1) b - a - 1
1
= ----------- is convergent
b - a - 1
∞ x^a
∴ ∫ --------- dx is convergent by comparison test
1 1 + x^b
1 x^a
Obviously , ∫ --------- dx is convergent
0 1 + x^b
∞ x^a
Hence , ∫ --------- dx is convergent
0 1 + x^b
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