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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New Hope is a borough in Bucks County, Pennsylvania, USA. The population was 2,528 at the 2010 census. The borough lies on the west bank of the Delaware River at its confluence with Aquetong Creek. The two-lane New Hope â Lambertville Bridge carries automobile and foot traffic across the Delaware to Lambertville, New Jersey on the east bank. New Hope's primary industry is tourism.