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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For number 5 , the problem is the picture on the screen. When I try to complete the square here I get (x-2)^2-13. Why is it (x-2)^2+1 in the answer key?
![**Problem 5:**
Describe a first step in integrating the function:
\[
\int \frac{10}{x^2 - 4x - 9} \, dx.
\]
**Explanation:**
The first step in integrating a rational function like this one is often to simplify and factor the expression in the denominator, if possible. Factoring the quadratic expression \(x^2 - 4x - 9\) might help break down the integral into partial fractions, making it easier to solve. Check if the quadratic can be factored into two binomials or use the quadratic formula to find its roots.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2F187377e6-0c1f-46a8-bc25-5cd414637814%2Fvy3qg9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
Describe a first step in integrating the function:
\[
\int \frac{10}{x^2 - 4x - 9} \, dx.
\]
**Explanation:**
The first step in integrating a rational function like this one is often to simplify and factor the expression in the denominator, if possible. Factoring the quadratic expression \(x^2 - 4x - 9\) might help break down the integral into partial fractions, making it easier to solve. Check if the quadratic can be factored into two binomials or use the quadratic formula to find its roots.
![### Completing the Square in Integration
#### Problem 8.1.5
**Task:**
Complete the square in the denominator to solve the integral:
\[ \int \frac{10}{(x-2)^2 + 1} \, dx. \]
**Notes:**
- The handwritten note suggests checking if \((x-2)^2 - 13\) is relevant to solving the problem.
- There is an annotation with a question mark indicating uncertainty about the completed square form.
- The process involves completing the square to simplify the integrand for easier integration.
**Discussion:**
Completing the square is a technique used to simplify quadratic expressions. In this case, it helps transform the integrand into a form that resembles the standard inverse tangent (arctan) integration formula.
By rearranging and simplifying the expression in the denominator, you ensure it takes a recognizable form, often aiding in the application of standard integration techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61b75115-d70f-4fe2-af93-2076876ad69a%2F187377e6-0c1f-46a8-bc25-5cd414637814%2Fgryzzw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Completing the Square in Integration
#### Problem 8.1.5
**Task:**
Complete the square in the denominator to solve the integral:
\[ \int \frac{10}{(x-2)^2 + 1} \, dx. \]
**Notes:**
- The handwritten note suggests checking if \((x-2)^2 - 13\) is relevant to solving the problem.
- There is an annotation with a question mark indicating uncertainty about the completed square form.
- The process involves completing the square to simplify the integrand for easier integration.
**Discussion:**
Completing the square is a technique used to simplify quadratic expressions. In this case, it helps transform the integrand into a form that resembles the standard inverse tangent (arctan) integration formula.
By rearranging and simplifying the expression in the denominator, you ensure it takes a recognizable form, often aiding in the application of standard integration techniques.
Expert Solution
Step 1
This problem is related to the integration. By looking into this problem we can see first of all we will have to simplify the denominator.
We will use the square rule for doing the same.
(a+b)^2 = a^2 + 2ab +b^2
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