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5.c – The Fundamental Theorem of Calculus and Definite Integrals

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Examples The definite integral of f (x) from x = a to x = b is denoted f(x) is called the integrand, a the lower limit of integration, and b the upper limit of integration.

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The First Fundamental Theorem of Calculus

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4 Basic Properties of the Indefinite Integral Let a, b, and c be constants and f and g be continuous functions on [a, b].

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Examples Evaluate:

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6 Definite Integrals With The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Properties of Odd and Even: Suppose f is continuous on [– a, a].

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7 Examples - Evaluate Evaluate by changing your limits of integration to values that are in terms of u.

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First Fundamental Theorem of Calculus (Alternate Definition) We’ve shown that represents the general antiderivative of f with respect to x. It follows that the derivative of the antiderivative should return the original function (that is, the integrand). t is called a dummy variable. Upper Limit Must Be Variable Part Lower Limit Must Be Numerical Part

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Examples Evaluate the following derivatives.

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Examples Let u is some function of x. Use WolframAlpha to determine the following:

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First Fundamental Theorem of (Generalized) If f is continuous on [a, b] and u is an unknown, differentiable function of x, then

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Examples Evaluate

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