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
![**Using the Fundamental Theorem of Calculus to Evaluate a Definite Integral**
Evaluate the definite integral using the Fundamental Theorem of Calculus and the antiderivative found in Step 2:
\[
\int_{0}^{1} \frac{x^2 - 9}{1 + x^2} \, dx = \int_{0}^{1} \left( \frac{x^2 + 1 - 10}{1 + x^2} \right) \, dx
\]
\[
= \int_{0}^{1} \left( 1 - \frac{10}{1 + x^2} \right) \, dx
\]
\[
= \left( x - 10 \arctan(x) \right) \bigg|_{0}^{1}
\]
\[
= \left( 1 - 10 \arctan(1) \right) - \left( \Box - 10 \arctan(\Box) \right)
\]
Finally, substitute and calculate the values to complete the evaluation:
\[
= \Box
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe1f37d79-53bb-43ed-93c0-29d608b5ceba%2F1bdc549c-c358-4200-8de3-dc9263608802%2F54w4nx_processed.png&w=3840&q=75)
Transcribed Image Text:**Using the Fundamental Theorem of Calculus to Evaluate a Definite Integral**
Evaluate the definite integral using the Fundamental Theorem of Calculus and the antiderivative found in Step 2:
\[
\int_{0}^{1} \frac{x^2 - 9}{1 + x^2} \, dx = \int_{0}^{1} \left( \frac{x^2 + 1 - 10}{1 + x^2} \right) \, dx
\]
\[
= \int_{0}^{1} \left( 1 - \frac{10}{1 + x^2} \right) \, dx
\]
\[
= \left( x - 10 \arctan(x) \right) \bigg|_{0}^{1}
\]
\[
= \left( 1 - 10 \arctan(1) \right) - \left( \Box - 10 \arctan(\Box) \right)
\]
Finally, substitute and calculate the values to complete the evaluation:
\[
= \Box
\]
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