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Chapter 4 | Applications of Derivatives
∫ 2 xdx = x 2 + C . The collection of all functions of the form x 2 + C , where C is any real number, is known as the family of antiderivatives of 2 x . Figure 4.85 shows a graph of this family of antiderivatives.
Figure 4.85 The family of antiderivatives of 2 x consists of all functions of the form x 2 + C , where C is any real number.
For some functions, evaluating indefinite integrals follows directly from properties of derivatives. For example, for n ≠−1, ∫ x n dx = x n +1 n +1 + C , which comes directly from d dx ⎛ ⎝ x n +1 n +1 ⎞ ⎠ = ( n +1) x n n +1 = x n . This fact is known as the power rule for integrals .
Theorem 4.15: Power Rule for Integrals For n ≠−1,
n +1 n +1 +
∫ x n dx = x
C .
Evaluating indefinite integrals for some other functions is also a straightforward calculation. The following table lists the indefinite integrals for several common functions. A more complete list appears in Appendix B .
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