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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Question
![**Problem Statement:**
Find the derivative of the function:
\[
g(x) = \int_{3x}^{6x} \frac{u^2 - 1}{u^2 + 1} \, du
\]
**Hint:**
To solve this problem, consider the following identity:
\[
\int_{3x}^{6x} f(u) \, du = \int_{3x}^{0} f(u) \, du + \int_{0}^{6x} f(u) \, du
\]
Here, \( f(u) = \frac{u^2 - 1}{u^2 + 1} \).
**Explanation of Terms:**
- The expression inside the integral, \( \frac{u^2 - 1}{u^2 + 1} \), is the function to be integrated with respect to \( u \).
- The limits of integration are functions of \( x \) (namely, \( 3x \) and \( 6x \)), which is important when applying the Leibniz rule to find the derivative of the integral with respect to \( x \).
- The hint provides a method to split the integral over a variable limit, which can be useful for further simplification or evaluation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F11934756-aeef-40fb-829b-e2c209a84cd6%2Fee1cee3a-7000-41cb-99aa-bb42d63df3fb%2F2ib8aw_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the function:
\[
g(x) = \int_{3x}^{6x} \frac{u^2 - 1}{u^2 + 1} \, du
\]
**Hint:**
To solve this problem, consider the following identity:
\[
\int_{3x}^{6x} f(u) \, du = \int_{3x}^{0} f(u) \, du + \int_{0}^{6x} f(u) \, du
\]
Here, \( f(u) = \frac{u^2 - 1}{u^2 + 1} \).
**Explanation of Terms:**
- The expression inside the integral, \( \frac{u^2 - 1}{u^2 + 1} \), is the function to be integrated with respect to \( u \).
- The limits of integration are functions of \( x \) (namely, \( 3x \) and \( 6x \)), which is important when applying the Leibniz rule to find the derivative of the integral with respect to \( x \).
- The hint provides a method to split the integral over a variable limit, which can be useful for further simplification or evaluation.
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