(相当于令 t=φ(x)t = \varphi(x)t=φ(x))
∫abf[φ(x)]φ′(x)dx=∫abf[φ(x)]d[φ(x)]=F[φ(x)]∣ab \int_a^b f[\varphi(x)] \varphi'(x) dx = \int_a^b f[\varphi(x)] d[\varphi(x)] = F[\varphi(x)]|_a^b ∫abf[φ(x)]φ′(x)dx=∫abf[φ(x)]d[φ(x)]=F[φ(x)]∣ab
令 x=φ(t)x = \varphi(t)x=φ(t),
∫abf(x)dx=∫φ−1(a)φ−1(b)f[φ(t)]φ′(t)dt \int_a^b f(x) dx = \int_{\varphi^{-1} (a)}^{\varphi^{-1} (b)} f[\varphi(t)] \varphi'(t) dt ∫abf(x)dx=∫φ−1(a)φ−1(b)f[φ(t)]φ′(t)dt
提示
注意: 不需要像不定积分那样回代反函数, 只需要将 a,ba, ba,b 对应的 ttt 带入计算即可 但记得换积分上下限
∫abu(x)d[v(x)]=u(x)d(x)∣ab−∫abv(x)d[u(x)] \int_a^b u(x) d[v(x)] = u(x) d(x)|_a^b - \int_a^b v(x) d[u(x)] ∫abu(x)d[v(x)]=u(x)d(x)∣ab−∫abv(x)d[u(x)]
∫abudv=uv∣ab−∫abvdu \int_a^b u dv = uv|_a^b - \int_a^b v du ∫abudv=uv∣ab−∫abvdu
注意
注意: uvuvuv 需要相减
