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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Spencer County, IN Societies and Family History Resources. 1  Spencer County Historian, David Ranger, 210 Walnut St., Rockport, IN 47635-1357; Home Phone: (812) 649-2582. 2  Spencer County Historical Society, 210 Walnut St., Rockport, IN 47635-1357; Phone: (812) 649-4866. 3  Indiana State Archives, 6440 E. 30th St., Indianapolis, IN 46219; (317) 591-5222.