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Integration Rules
THEOREM 2 Let u and v be functions of x whose domains are an open interval I and suppose du and dv exist for every x in I.
Discussion The Power Rule gives the integral of u^{r} when r 1, while Rule (viii) gives the integral of u^{r} when r = 1. When we put u =f(x) and v = g(x), the Constant and Sum Rules take the form Constant Rule c f(x) dx = c f(x) dx. Sum Rule (f (x) + g(x)) dx = f (x) dx + g(x) dx. In the Constant and Sum Rules we are multiplying a family of functions by a constant and adding two families of functions. If we do either of these two things to families of functions differing only by a constant, we get another family of functions differing only by a constant. For example, 7(3x^{4} + C) = 21x^{4} + 7C = 21x^{4} + C' is the family of all functions equal to 21x^{4} plus a constant. Similarly, (3√x + C) + (5x  √x + D) = 5x + 2√x + (C + D) = 5x + 2√x + C is the family of all functions equal to 5x + 2√x plus a constant. PROOF OF THEOREM 2
Rules (v)(viii) are similar. Only the last formula, (viii), requires an explanation. The absolute value in ln u comes about by combining the two cases u > 0 and u < 0. When u > 0, u = u and d(ln u) = d(ln u) = du. When w < 0, ln u is undefined, but u =  u and ln u = ln (u). Thus d(lnu) = d(ln(u)) = 1/u d(u) = 1/u du. Thus, in both cases, when u 0,


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