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Unformatted text preview: 1 Indefinite Integrals The fundamental theorem of calculus shows just how important antiderivatives are. Since we’ll be using them so frequently from now on, we introduce a notation for them. Actually, the FTC gives us an intuitive notation for them. Definition 1. R f ( x ) dx is an antiderivative of f ( x ). This is called the indefinite integral. *Do a few examples It is incredibly important to remember the difference between a definite and an indefinite integral. R b a f ( x ) dx is a number and x shouldn’t appear anywhere. R f ( x ) dx is a family of functions and should be nothing but x ’s. Here’s a quick reminder of antidifferentiation formulas. *Put up a table Example 1.1. Z (18 x 5- 3 sec x tan x ) dx Example 1.2. Z sin θ cos 2 θ dx *Do a few examples of definite integrals. Finally, recall how I was talking about how the definite integral is a net area or a displacement. Well, another way to state the fundamental theorem of calculus is that the integral of the rate of change is the net change. *Talk about displacement and then some other examples. Example 1.3. v ( t ) = t 2- t- 6. Find the displacement and the distance travelled....
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