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Calculus: Differentiating Integral with Functions in Both Bounds

Learn how to differentiate an integral with functions in both bounds using the additive property and the chain rule. Apply the First Fundamental Theorem of Calculus to find the derivative.

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Calculus: Differentiating Integral with Functions in Both Bounds

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  1. Example 9 Find Solution We only know how to differentiate a an integral with a function of x in its upper bound. However, the given integral has functions of x in both of its bounds. Therefore, begin by using the additive property to write the given integral as the sum of two integrals, each with one constant bound: Before differentiating, interchange the bounds of the first integral: Let By the chain rule:

  2. By the First Fundamental Theorem of Calculus, the derivative F /(w) is obtained by replacing t by w in the integrand Apply this formula with w=u and with w=v:

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