推 znmkhxrw :第一題少個負號~~ 03/02 11:41
※ 引述《iddee ()》之銘言：
: oo sin^2(8x) - sin^2(2x)
: 1. ∫ --------------------- dx (Hint: Fubini Thm)
: 0 x
sketch: Since (sinx)^2 = (1-cos2x)/2 , let a > 0
∞ cos16x - cos4x
we have lim ∫ --------------- dx
a→0 a 2x
∞ cos16x ∞ cos4x
= lim [∫ ------- dx -∫------- dx]
a→0 a 2x a 2x
∞ cost ∞ cost
= lim [∫ ------- dt -∫------ dt]
a→0 16a 2t 4a 2t
4a cost
= lim [∫ ------- dt ] ( let t = ax)
a→0 16a 2t
4 cosax
= lim [∫ ------- dx ]
a→0 16 2x
4 cosax
= ∫ lim ------- dx
16 a→0 2x
: 1 (x-1)x^k
: 2. ∫ ---------- dx for k ≧ 0
: 0 ln(x)
1 x^(k+1) - x^k 1 k+1
∫ --------------- dx = ∫ ∫ x^y dy dx
0 lnx 0 k
k+1 1 k+1 x^(y+1) ｜1
= ∫ ∫ x^y dx dy = ∫ ---------- ｜ dy
k 0 k y+1 ｜0
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