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高数不定积分设f(x)的原函数为F(x)>0,且F(0)=0,当x≥0时,f(x)F(x)=sin^2(2x),求f(x)
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高数不定积分
设f(x)的原函数为F(x)>0,且F(0)=0,当x≥0时,f(x)F(x)=sin^2(2x),求f(x)
设f(x)的原函数为F(x)>0,且F(0)=0,当x≥0时,f(x)F(x)=sin^2(2x),求f(x)
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答案和解析
∫ ƒ(x) dx = F(x)
F'(x) = ƒ(x)
ƒ(x)F(x) = sin²2x = (1 - cos4x)/2,两边积分
∫ ƒ(x)F(x) dx = ∫ [1/2 - (1/2)cos4x] dx
∫ F(x) d[F(x)] = ∫ [1/2 - (1/2)cos4x] dx
(1/2)[F(x)]² = x/2 - (1/8)sin4x + C
[F(x)]² = x - (1/4)sin4x + C
F(x) = √[x - (1/4)sin4x + C]
F(0) = 0 ==> √[0 - 0 + C] = 0 ==> C = 0
F(x) = √[x - (1/4)sin4x]
F'(x) = 2sin²2x/√(4x - sin4x) = ƒ(x)
F'(x) = ƒ(x)
ƒ(x)F(x) = sin²2x = (1 - cos4x)/2,两边积分
∫ ƒ(x)F(x) dx = ∫ [1/2 - (1/2)cos4x] dx
∫ F(x) d[F(x)] = ∫ [1/2 - (1/2)cos4x] dx
(1/2)[F(x)]² = x/2 - (1/8)sin4x + C
[F(x)]² = x - (1/4)sin4x + C
F(x) = √[x - (1/4)sin4x + C]
F(0) = 0 ==> √[0 - 0 + C] = 0 ==> C = 0
F(x) = √[x - (1/4)sin4x]
F'(x) = 2sin²2x/√(4x - sin4x) = ƒ(x)
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