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The Chain Rule and Integration by Substitution Recall: The chain rule for derivatives allows us to differentiate a composition of functions: derivative [ f (g(x))]' = f '(g(x))g'(x) antiderivative € The Chain Rule and Integration by Substitution Suppose we have an integral of the form where F'= f. ∫ f (g(x))g'(x)dx composition of derivative of F is an antiderivative of f functions Inside function € € Then, by reversing the chain rule for derivatives, we have ∫ f(g(x))g'(x)dx = F(g(x))+C. integrand is the result of differentiating a composition of functions € Example 2x+5 Integrate ∫ x2 +5x−7 dx € Integration by Substitution Algorithm: u=g(x) g(x) 1. Let where is the part causing g'(x) problems and cancels the remaining x terms in the integrand. € € u=g(x) 2. Substitute and into the € du=g'(x)dx integral to obtain an equivalent (easier!) integral all in terms of u. € € ∫ f (g(x))g'(x)dx = ∫ f (u)du €
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