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HAVING FUN WITH ANTI-DERIVATIVES Actually, the first thing we are going to do is to demonstrate a neat formula, namely the statement: Here is why: let be an anti-derivative of. Then, by the FTC, version 2 (we called it a corol- lary !)

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(Don’t be upset by Sam and Sue, we will give them proper names later!) So Let’s give Sam and Sue their proper names, And re-write the formula above as

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(ready?) But now, by the chain rule and

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The moral of this story is: 1 When taking the derivative of an integral 2 Use the FTC (evaluate the integrand properly) 3 But don’t forget the chain rule ! Here is an example:

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Another example: Do a few more from the textbook.

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Anti-derivative is an ugly word ! Using the con- nection between anti-derivatives and integrals provided by the FTC we write or more conventionally for any one anti-derivative of and call it the indefinite integral of a much better name! Now we get a few bricks and some mortar. We start with the mortar (rules)

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(very few) Now the bricks (depends on your memory !)

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and from trigonometry Any more ?

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4.1 Antiderivatives and Indefinite Integration Definition of Antiderivative: A function F is called an antiderivative of the function f if for every x.

4.1 Antiderivatives and Indefinite Integration Definition of Antiderivative: A function F is called an antiderivative of the function f if for every x.

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