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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Hope Valley is a village and census-designated place (CDP) in the town of Hopkinton in Washington County, Rhode Island, United States. The population of the CDP was 1,649 at the 2000 census. Hope Valley is the largest village in Hopkinton and the town's principal commercial center. While the village of Hope Valley is located in Hopkinton, its zip code, 02832, extends into the neighboring town of Richmond.