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Why is $\int_{-1}^1 x =\frac{x^2}{2}$ rather than $x^2$?

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    The integral of $x$ from -1 to 1 is neither of those! Remember that a definite integral (with limits of integration) is a *number* (think area under the curve) not a function. If you leave off the limits of integration, then you're talking about the *antiderivative*, and your question makes more sense.2010-12-17

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