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 township is about 70 miles, along Lake Ontario, East of Toronto. (Hope township is now the town of Port Hope in Northumberland county.) In 1856 or before, they moved to Lindsay, Ontario. Lindsay is in Ops township in Victoria County, Ontario, which is the county immediately north of Durham county. Lindsay is about 50 miles to the north, and slightly west, of Hope township. (On January 1, 2001, Victoria county, Ontario, became the City of Kawartha Lakes, a single-tier municipality.)