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Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate. VES +1 dt d 10 dx x3

Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate. VES +1 dt d 10 dx x3

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Topic: Application of the Fundamental Theorem of Calculus** **Objective:** Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate the expression. **Expression to Evaluate:** \[ \frac{d}{dx} \left[ \int_{x^3}^{10} \sqrt{t^6 + 1} \, dt \right] \] **Explanation:** This problem requires the use of the Fundamental Theorem of Calculus, which connects differentiation and integration. Part 1 of this theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then: \[ \frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) \, dt \right] = f(g(x)) \cdot g'(x) \] In this case, we will apply this theorem along with the Chain Rule to evaluate the integral, treating the variable limit of integration as a function of \( x \).
Transcribed Image Text:**Topic: Application of the Fundamental Theorem of Calculus** **Objective:** Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate the expression. **Expression to Evaluate:** \[ \frac{d}{dx} \left[ \int_{x^3}^{10} \sqrt{t^6 + 1} \, dt \right] \] **Explanation:** This problem requires the use of the Fundamental Theorem of Calculus, which connects differentiation and integration. Part 1 of this theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then: \[ \frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) \, dt \right] = f(g(x)) \cdot g'(x) \] In this case, we will apply this theorem along with the Chain Rule to evaluate the integral, treating the variable limit of integration as a function of \( x \).
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