Integration by substitution: Difference between revisions

Revision as of 20:18, 15 August 2020

Revision as of 14:09, 27 October 2020

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:<math>\int_{\varphi(a)}^{\varphi(b)} f(u)\,du</math>

where {{math|''u'' {{=}} ''φ''(''x'')}} in fact exist, and it remains to show that they are equal.

Since {{math|''f''}} is continuous, it has an [[antiderivative]] {{math|''F''}}. The [[function composition|composite function]] {{math|''F'' ∘ ''φ''}} is then defined. Since {{math|''φ''}} is differentiable, combining the [[chain rule]] and the definition of an antiderivative gives