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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Noble County, Indiana. Noble County is a county located in the U.S. state of Indiana. As of 2010, the population was 47,536. The county seat is Albion. The county is divided into 13 townships which provide local services.