※ 引述《BIEGABOY (BIEGABOY)》之銘言：
: 1.
: 1
: 若∫ f(x)dx = 1/5 , f(x)-1/5 = 0 有實根
: 0
1/4 ,0≦x≦1/2
(f(x) should be a continous function,or let f(x)={ will be a
3/20,1/2<x≦1
counterexample)
suppose not
then either f(x)-1/5 > 0 or f(x)-1/5 < 0 will hold for all real x
we have:
1 1 1 1
∫ f(x)dx > ∫ 1/5 dx =1/5 or ∫ f(x)dx < ∫ 1/5 dx =1/5
0 0 0 0
contradiction
: 2.
: 100 100
: ∫ √(100-x)
: 0 ------------------- dx
: 100 100
: √(100-x) + √x
Let u=100-x then du=-dx
100 (100-x)^(1/100) 0 u^(1/100)
I = ∫----------------------------dx = ∫ ------------------------------(-du)
0 (100-x)^(1/100) + x^(1/100) 100 u^(1/100) + (100-u)^(1/100)
100 u^(1/100)
= ∫ ------------------------------du
0 (100-u)^(1/100) + u^(1/100)
100 x^(1/100)
= ∫ ------------------------------dx
0 (100-x)^(1/100) + x^(1/100)
and
100 (100-u)^(1/100) + u^(1/100)
∫ ------------------------------du = 100
0 (100-u)^(1/100) + u^(1/100)
so
I = 50
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