Unformatted text preview: namely Z 2 1 x 2 dx = 2 3 31 3 3 = 7 3 . We use Z x 2 dx to denote the antiderivative of x 2 , also called an indeﬁnite integral . So this is evaluated as Z x 2 dx = x 3 3 + C. It is customary to include the constant C to indicate that there are really an inﬁnite number of antiderivatives. We do not need this C to compute deﬁnite integrals, but in other circumstances we will need to remember that the C is there, so it is best to get into the habit of writing the C . When we compute a deﬁnite integral, we ﬁrst ﬁnd an antiderivative and then substitute. It is convenient to ﬁrst display the antiderivative and then do the substitution; we need a notation indicating that the substitution is yet to be done. A typical solution would look like this: Z 2 1 x 2 dx = x 3 3 ﬂ ﬂ ﬂ ﬂ 2 1 = 2 3 31 3 3 = 7 3 . The vertical line with subscript and superscript is used to indicate the operation “substitute and subtract” that is needed to ﬁnish the evaluation....
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 Spring '07
 JonathanRogawski
 Math, Calculus, Derivative, Fundamental Theorem Of Calculus

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