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Mathematics-Online lexicon:

Substitution

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
From the chain rule

$\displaystyle \frac{d}{dx} F(g(x))=f(g(x)) g'(x),\quad f=F',$

follows by integration

$\displaystyle \int f(g(x)) g'(x) dx = F(g(x)) +c.$

According to definite integrals holds

$\displaystyle \int_a^b f(g(x)) g'(x) dx = F(g(b))-F(g(a)) = \int_{g(a)}^{g(b)}f(y) dy. $

By means of differentials one write this formula in the form

$\displaystyle \int_a^b f(y) \frac{dy}{dx}dx = \int_{g(a)}^{g(b)} f(y)dy. $

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  automatically generated 4/ 8/2008