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 Township of Indiana was incorporated in 1805 and is 17+ square miles in size. It was the fourth township formed in Allegheny County north of the Ohio and Allegheny Rivers. The Township has approximately 30 miles of its own roads and there are more than 22 miles of State and County roads within the municipality. We have approximately 7,000 residents in one of the most rapidly developing communities that comprise the Fox Chapel Area School District.