∫f[φ(x)]φ′(x)dx=F(φ(x))+C \int f[\varphi(x)] \varphi'(x) dx = F(\varphi(x)) + C ∫f[φ(x)]φ′(x)dx=F(φ(x))+C
即: 令 t=φ(x)t = \varphi(x)t=φ(x), 则 dt=φ′(x)dxdt = \varphi'(x) dxdt=φ′(x)dx, 原式 =∫f(t)dt=F(t)+C= \int f(t) dt = F(t) + C=∫f(t)dt=F(t)+C
∫f(x)dx=∫f[φ(t)]φ′(t)dt=F(t)+C=F(φ−1(x))+C \int f(x) dx = \int f[\varphi(t)] \varphi'(t) dt = F(t) + C = F(\varphi^{-1}(x)) + C ∫f(x)dx=∫f[φ(t)]φ′(t)dt=F(t)+C=F(φ−1(x))+C
即: 令 x=φ(t)x = \varphi(t)x=φ(t), 则 dx=φ′(t)dtdx = \varphi'(t) dtdx=φ′(t)dt, ∫f(x)dx=∫f[φ(t)]φ′(t)dt\int f(x) dx = \int f[\varphi(t)] \varphi'(t) dt∫f(x)dx=∫f[φ(t)]φ′(t)dt
注意
注意换元求完积分后记得换回 x 来