ACOW INDEFINITE INTEGRALS MODULE
Updated 5/28/2016
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Activity A1- Antiderivatives
When the derivative, f ( x) , of a function, F ( x) , is known, we say that F ( x) is the
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antiderivative of f ( x) . For example, given f ( x)  x 2  2 x  4 , we can determine if
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F ( x)  x  x 2  4 x  7 is the antiderivative of f ( x ) by finding if F '( x)  f ( x) . Since
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F '( x)  x 2  2 x  4  f ( x) , we conclude f ( x ) is the antiderivative of F ( x) .
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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.
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x  x 2  4 x  7 , G ( x)  x 3  x 2  4 x  10 , and
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H ( x)  x 3  x 2  4 x  4 are all antiderivatives of f ( x)  x 2  2 x  4 . Why? Recall,
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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
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f ( x)  x 2  2 x  4 is F ( x)  x3  x 2  4 x  C where C is any constant.
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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
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F ( x) 
This is an undefined function because
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x
0
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is an undefined number.
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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) 
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x7
6. f ( x)  5 x3