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ACOW INDEFINITE INTEGRALS MODULE Updated 5/28/2016 Page 1 of 2 Activity A1- Antiderivatives When the derivative, f ( x) , of a function, F ( x) , is known, we say that F ( x) is the 3 antiderivative of f ( x) . For example, given f ( x) x 2 2 x 4 , we can determine if 2 1 3 F ( x) x x 2 4 x 7 is the antiderivative of f ( x ) by finding if F '( x) f ( x) . Since 2 3 F '( x) x 2 2 x 4 f ( x) , we conclude f ( x ) is the antiderivative of F ( x) . 2 1. Which of the following is the antiderivative of f ( x) 3x 2 4 x 8 ? (a) F ( x) x 3 4 x 2 8 (b) F ( x) 3x 3 2 x 2 8 x 7 (c) F ( x) 3 x 3 4 x 2 8 x (d) F ( x) x 3 2 x 2 8 x 5 (e) None of the above. 1 3 1 x x 2 4 x 7 , G ( x) x 3 x 2 4 x 10 , and 2 2 1 3 H ( x) x 3 x 2 4 x 4 are all antiderivatives of f ( x) x 2 2 x 4 . Why? Recall, 2 2 that the derivative of a constant is zero and thus we “lose” the constant in the derivative process. Consequently, when we try to find the antiderivative of a function, we do not know the actual value of the constant. Thus we say the most general antiderivative of 3 1 f ( x) x 2 2 x 4 is F ( x) x3 x 2 4 x C where C is any constant. 2 2 It is important to realize F ( x) 2. Which of the following is the most general antiderivative of f ( x) 4 x3 3x 2 7 ? (a) F ( x) x 4 x 3 7 x C (b) F ( x) x 4 x 3 7 x (c) F ( x) x 4 x 3 7 x 2 (d) F ( x) x 4 x 3 7 x 7 When we are given a function and are asked to find it’s antiderivative, we must use the Power Rule for Antiderivatives(RS2). Notice the Power Rule for Antiderivatives states n 1 . This is necessary because if n 1 , then we would have the function f ( x) x 1 . Trying to apply the power rule would give ACOW INDEFINITE INTEGRALS MODULE Updated 5/28/2016 Page 2 of 2 F ( x) This is an undefined function because 1 0 x 0 1 is an undefined number. 0 Use the Power Rule for Antiderivatives to find the most general antiderivative of the following functions. 3. f ( x) x5 . 4. f ( x) x . 5. f ( x) 1 x7 6. f ( x) 5 x3