Integration by substitution can be derived from the fundamental theorem of calculus as follows. Let Æ and Ï be two functions satisfying the above hypothesis that Æ is continuous on I and Ïâ² is continuous on the closed interval [a,b] . Then the function Æ(Ï(t))Ïâ²(t) is also continuous on [a,b] . Hence the integrals.
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The Hopesâ son Anthony died in 2004 at age 63. Mrs. Hope is survived by two daughters, Linda Hope, a television producer, of Toluca Lake, and Nora Hope; another son, William, of Oakland, Calif.; three grandchildren; and a great-grandson. Mrs. Hope described herself to interviewers as grateful for her family life.