note9 - Math021, week 9 Antiderivatives Definition 9.1 If f...

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Unformatted text preview: Math021, week 9 Antiderivatives Definition 9.1 If f and F are functions such that F = f , F is called an antiderivative of f . Remark 9.2 If F is an antiderivative of f , so is F + C for any constant function C . Definition 9.3 If F is an antiderivative (or indefinite integral) of f , we write Z f ( x ) dx = F ( x ) + C. Remark 9.4 We get formula right away from the formula for derivatives. For instance, • R x k dx = x k +1 k +1 + C if k 6 =- 1 • R sin xdx =- cos x + C • R cos xdx = sin x + C • R sec 2 xdx = tan x + C • R csc 2 xdx =- cot x + C • R tan x sec xdx = sec x + C • R cot x csc xdx =- csc x + C • R e x dx = e x + C • R ln xdx = 1 x + C Lemma 9.5 If f and g are functions and a is a number, 1. R ( f + g )( x ) dx = R f ( x ) dx + R g ( x ) dx 2. R ( af )( x ) dx = a R f ( x ) dx 1 proof of 1) Suppose that F = f and G = g . Then, ( F + G ) = F + G = f + g . Hence, an antiderivative of f + g is Z ( f + g )( x ) dx = F + G = Z f ( x ) dx + Z g ( x ) dx. Example 9.6 An antiderivative of f ( x ) = x 2 + 2 x + 3sin x is F ( x ) = 1 3 x 3 + 2ln x- 3cos x + C....
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note9 - Math021, week 9 Antiderivatives Definition 9.1 If f...

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