作者:因情语写
链接:https://www.proprogrammar.com/article/140
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定理1 假设函数f(x)在区间[a, b]上连续,函数x=φ(t)满足条件:
(1)φ(α) = a, φ(β) = b
(2)φ(t)在[α, β](或[β, α])上具有连续导数,且其值域R = [a, b],则有
∫[a, b]f(x)dx = ∫[α, β]f[φ(t)]φ'(t)dt
注:
1、若f(x)在[-a, a]上连续且为偶函数,则∫[-a, a]f(x)dx = 2∫[0, a]f(x)dx
2、若f(x)在[-a, a]上连续且为奇函数,则∫[-a, a]f(x)dx = 0
3、设f(x)是连续的周期函数,周期为T,则有
(1)∫[a,a+T]f(x)dx = ∫[0, T]f(x)dx
(2)∫[a,a+nT]f(x)dx = n∫[0, T]f(x)dx (n∈N)
∫[a, b]uv'dx = ∫[a, b]udv = [uv][a, b] - ∫[a, b]vu'dx
注:与不定积分类似
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