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Chapter 4 | Applications of Derivatives
Theorem 4.14: General Form of an Antiderivative Let F be an antiderivative of f over an interval I . Then, i. for each constant C , the function F ( x )+ C is also an antiderivative of f over I ; ii. if G is an antiderivative of f over I , there is a constant C for which G ( x ) = F ( x )+ C over I . In other words, the most general form of the antiderivative of f over I is F ( x )+ C .
We use this fact and our knowledge of derivatives to find all the antiderivatives for several functions.
Example 4.50 Finding Antiderivatives
For each of the following functions, find all antiderivatives. a. f ( x ) =3 x 2 b. f ( x ) = 1 x c. f ( x ) =cos x d. f ( x ) = e x
Solution
a. Because
⎛ ⎝ x 3
⎞ ⎠ =3 x 2
d dx
then F ( x ) = x 3 is an antiderivative of 3 x 2 . Therefore, every antiderivative of 3 x 2 is of the form x 3 + C for some constant C , and every function of the form x 3 + C is an antiderivative of 3 x 2 . b. Let f ( x ) = ln| x |. For x >0, f ( x ) = ln( x ) and d dx (ln x ) = 1 x . For x <0, f ( x ) = ln(− x ) and d dx ⎛ ⎝ ln(− x ) ⎞ ⎠ = − 1 − x = 1 x .
Therefore,
d dx (ln|
x |) = 1 x .
Thus, F ( x ) = ln| x | is an antiderivative of 1 x is of the form ln| x | + C for some constant C and every function of the form ln| x | + C is an antiderivative of 1 x . c. We have x . Therefore, every antiderivative of 1
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