The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages. |
Home Integral Indefinite Integrals Integration Rules | |||||||||||||||||||||||||||||||||
See also: Integral of Products | |||||||||||||||||||||||||||||||||
Search the VIAS Library | Index | |||||||||||||||||||||||||||||||||
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 ur when r -1, while Rule (viii) gives the integral of ur 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(3x4 + C) = 21x4 + 7C = 21x4 + C' is the family of all functions equal to 21x4 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(ln|u|) = d(ln(-u)) = -1/u d(-u) = -1/u du. Thus, in both cases, when u 0,
|
|||||||||||||||||||||||||||||||||
Home Integral Indefinite Integrals Integration Rules |