Module 8 the antiderivative substitution rule
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Integration by substitution allows solving integrals of the form ∫ f(g(x)) g'(x) dx by making the substitution u = g(x) and du = g'(x) dx. This transforms the integral into the simpler form ∫ f(u) du. For example, the integral ∫ cos (x2) 2x dx can be solved using this method by letting f=cos, g=x2, and g'=2x.
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Module 8 the antiderivative substitution rule
- 1. "Integration by Substitution" (also called "u-Substitution" or "The Reverse Chain Rule") is a method to find an integral, but only when it can be set up in a special way. The first and most vital step is to be able to write our integral in this form: ∫ 𝒇(𝒈(𝒙)) 𝒈′(𝒙) 𝒅𝒙 Note that we have 𝑔(𝑥) and its derivative 𝑔′(𝑥) EXAMPLE 1: ∫ 𝒄𝒐𝒔 (𝒙𝟐 ) 𝟐𝒙 𝒅𝒙 Here 𝑓 = 𝑐𝑜𝑠, and we have 𝑔 = 𝑥2 and its derivative 2𝑥. This integral is good to go! When our integral is set up like that, we can do this substitution: ∫ 𝑓(𝒈(𝒙)) 𝒈′(𝒙) 𝒅𝒙 ∫ 𝑓(𝒖) 𝒅𝒖 EXAMPLE 2: