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Integration by Substitution

Integration by Substitution - Rule for Integration If g is...

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Integration by Substitution Section 4.5
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Thm. 4.12 Antidifferentiation of a Composite Function Pattern Recognition: Change of Variables - u- substitution: If u = g(x) then du = g’(x)dx and f(g(x)) g’(x)dx = F(g(x)) + C f(u)du = F(u) + C (Ex. 1 and 2) (Ex. 4 and 6)
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Multiplying & Dividing by a Constant Many integrands contain the essential part (the variable part) of g’(x) but are missing a constant multiple. In such cases, you can multiply and divide by the necessary constant multiple. Look at Ex. 3 pg. 289
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Guidelines for Making a Change of Variables 1. Choose a substitution u = g(x) (Choose the inner part.) 2. Compute du = g’(x)dx. 3. Rewrite integral in terms of u. 4. Evaluate the integral in terms of u. 5. Replace u by g(x) . 6. Check answer by differentiating.
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Thm. 4.13 The General Power
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Unformatted text preview: Rule for Integration If g is a differentiable function of x , then If u = g(x), then [ g(x) ] n g’(x)dx = [ g(x) ] n+1 n + 1 + C u n du = u n+1 n + 1 + C, n = 1 n = 1 Thm. 4.14 Change of Variables for Definite Integrals If the function u = g(x) has a continuous derivative on the closed interval [ a, b ] and f is continuous on the range of g, then a b f(g(x)) g’(x) dx = g(b) g(a) f(u) du Thm. 4.15 Integration of Even and Odd Functions Let f be integrable on the closed interval [ a, -a ]. 1. If f is an even function, then- a a f(x) dx = 2 a f(x) dx 2. If f is an odd function, then - a a f(x) dx = 0 Joke Time How do you catch a unique rabbit? “Unique” up on him! How do you catch a tame rabbit? The “tame” way!...
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