變數變換(u-substitution)
Author: 藍邦偉  老師

$積分公式(3)u-substitution$

$或 \int{[f(u)\frac{du}{dx}] }dx=F(u)+c $

$ \frac{d}{dx} [F(u)]=\frac{d}{du}[F(u)]\cdot \frac{du}{dx}=f(u)\frac{du}{dx}$

$ ⇒\int{ f(u)du=F(u)+c} $

THM

Ex

$ \int{(x²+2x-1)^5(2x+2)dx } $

$ \int{ }\left [ f(u)\frac{du}{dx} \right ] dx=\int{ f(u)du} $

$ Pf $

$ 設F是f的一個反導函數，故 $

$ 若u為x的可微函數，則由連鎖律知 $

$ \frac{d}{du}[F(u)] =f(u)或\int{f(u)du }=F(u)+c$

$令u=x²+2x-1$
$⇒\frac{du}{dx}=2x+2$
$⇒du=(2x+2)dx$

$ 原式=\int{u^5du=\frac{1}{6}u^6+c$
           $          =\frac{1}{6}(x²+2x-1)^6+c $

$ 原式=\int{ (x²+2x-1)⁵d(x²+2x-1)} $
           $=\frac{1}{6}(x²+2x-1)^6+c$

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b = false
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