If {eq}\displaystyle F(x) = \int_{25}^{x} \frac{1}{t} \ dt {/eq}, then find {eq}\displaystyle F'(x) {/eq}


If {eq}\displaystyle F(x) = \int_{25}^{x} \frac{1}{t} \ dt {/eq}, then find {eq}\displaystyle F'(x) {/eq}

Leibniz's Rule:

It is a special method used to find the derivative of a definite integral without integration. If the definite integral is {eq}\displaystyle \int\limits_{A\left( x \right)}^{B\left( x \right)} {f\left( x \right)} dx {/eq}, then the following formula can be used to find the derivative of the definite integral.

{eq}\displaystyle \dfrac{d}{{dx}}\int\limits_{A\left( x \right)}^{B\left( x \right)} {f\left( x \right)} dx = f\left[ {B\left( x \right)} \right]\dfrac{d}{{dx}}B\left( x \right) - f\left[ {A\left( x \right)} \right]\dfrac{d}{{dx}}A\left( x \right) {/eq}

The above formula is known as Leibniz's formula.

Answer and Explanation: 1

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Given Data

  • The given definite integral is {eq}\displaystyle F(x) = \int\limits_{25}^x {\dfrac{1}{t}} \;dt {/eq}.

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Evaluating Definite Integrals Using the Fundamental Theorem


Chapter 16 / Lesson 2

In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.

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