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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LaPorte County is a county located in the U.S. state of Indiana. As of 2010, the population was 111,467. The county seat is the city of La Porte, and the largest city is Michigan City. This county is part of the Northwest Indiana and Michiana regions of the Chicago metropolitan area. The LaPorte County Courthouse is located in the county seat of La Porte and is listed on the National Register of Historic Places.