※ 引述《plover (>//////<)》之銘言：
: Suppose that f has derivative which is monotonic decreasing with
: f'(x) ≧ m > 0 on [a,b]. Show that
: b
: ∣∫ cosf(x) dx ∣ ≦2/m.
: a
Since f has derivative
∫ cosf(x) dx
[ cos f(x) ] * [ f'(x) ]
= ∫ ------------------------- dx
[ f'(x) ]
f is monotonic decreasing , so m ≧ 1/f`(b)
1/f` is increasing and cos u is continous , by 2nd M.V.T for
Riemann integrals
1 f(b)
----- ∫ [ cos u ] du
m f(c)
1
----- [ sin f(b) - sin f(c) ]
m
Hence, ∣∫ cosf(x) dx ∣
1
≦ ----- [ | sin f(b)| + | sin f(c) |] ≦ 2/m
m
Note: 其實有一個小地方沒說清楚 ... XD
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