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 had a population of 2,093 as of July 1, 2017. Hope ranks in the upper quartile for Population Density when compared to the other cities, towns and Census Designated Places (CDPs) in Indiana. See peer rankings below. The primary coordinate point for Hope is located at latitude 39.2993 and longitude -85.766 in Bartholomew County. The formal boundaries for the Town of Hope (see map below) encompass a land area of 0.95 sq. miles and a water area of 0 sq. miles.