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 town of Hope has a strong sense of history, as reflected in its Yellow Trail Museum, its location as the site of the Haw Creek branch of the Bartholomew County Historical Society, its annual Heritage Days celebration, and a number of other traditional celebrations.