Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online course: Preparatory Course Mathematics - Analysis - Integral Calculus

Substitution of Variables

[previous page] [next page] [table of contents][page 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. $

(temporary unavailable)

(temporary unavailable)
[previous page] [next page] [table of contents][page overview]
  automatically generated 1/9/2017