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Unformatted text preview: 5.4: Indefinite Integrals and the Net Change Theorem (Dated: September 23, 2011) INDEFINITE INTEGRALS Let f be a function on the interval [ a , b ]. Recall that an an- tiderivative of f is a function F such that F = f . The tradi- tional notaion to denote this statement is F ( x ) = R f ( x ) dx . In other word, Z f ( x ) dx = F ( x ) means F ( x ) = f ( x ). In this context, R f ( x ) dx is called an indefinite integral. Note that if Z f ( x ) dx = F ( x ), then so is Z f ( x ) dx = F ( x ) + C for any constant C . Hence, in fact, the indefinite integral R f ( x ) dx represents the entire family of antiderivatives of f . Example 1 (Exercise 5.4.2 in the text) Verify by differentia- tion that R x cos x dx = x sin x + cos x + C. Remark 1 For a table of some common indefinite integrals, refer to table 1 on p. 392 of the text. Example 2 (Exercises 5.4.6, 5.4.10, 5.4.11, 5.4.14, 5.4.18 in the text) Find the general indefinite integrals....
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