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# Day-7 - Math 1314 Day 7 Section 14.1 Antiderivatives So far...

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Math 1314 Day 7 Section 14.1 - Antiderivatives So far in this course, we have been interested in finding derivatives and in the applications of derivatives. In this chapter, we will look at the reverse process. Here we will be given the “answer” and we’ll have to find the “problem.” This process is generally called integration . We can use integration to solve a variety of problems. Antiderivatives Definition : A function F is an antiderivative of f on interval I if ( ) ( ) F x f x = for all x in I. The process of finding an antiderivative is called antidifferentiation or finding an indefinite integral . Example 1 : Determine if F is an antiderivative of f if 5 2 2 3 3 1 ) ( 2 3 + + + = x x x x F and . 2 3 ) ( 2 + + = x x x f

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Notation : We will use the integral sign to indicate integration (antidifferentiation). Problems will be written in the form + = . ) ( ) ( C x F dx x f This indicates that the indefinite integral of ) ( x f with respect to the variable x is C x F ) ( where ) ( x F is an antiderivative of f. Basic Rules Rule 1: The Indefinite Integral of a Constant + = C kx dx k Example 3 : dx 5 Rule 2: The Power Rule 1 , 1 1 - + + = + n C n x dx x n n Example 4 : dx x 4
Example 5 : 4 x dx Example 6 : 3 1 dx x

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Rule 3: The Indefinite Integral of a Constant Multiple of a Function = dx x f c dx x cf ) ( ) ( Example 7 : dx x 3 4 Example 8 : 4 2 dx x
Rule 4: The Sum (Difference) Rule [ ] ± = ± dx x g dx x f dx x g x f ) ( ) ( ) ( ) ( Example 9 : + + dx x x ) 1 5 2 ( 2 Rule 5: The Indefinite Integral of the Exponential Function + = C e dx e x x Example 10 : - dx x e x ) 4 5 ( 3

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Rule 6: The Indefinite Integral of the Function x x f 1 ) ( = + = 0 , | | ln 1 x C x dx x Example 11 : + - dx x x x 3 2 5 3
Example 12 : dx x x x x + - 3 2 5 4 3

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