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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![### Transcription for Educational Website
#### Calculus Question: Trigonometric Substitution in Integration
We start with the integral:
\[ \int \frac{9x^2}{(81 + x^2)^2} \, dx \]
Using the substitution \[ x = 9 \, \tan \theta \], we find:
\[ dx = 9 \, \sec^2 \theta \, d\theta \]
This transforms the integral into:
\[ \int \frac{9 (9 \tan \theta)^2 \cdot 9 \sec^2 \theta \, d\theta}{(81 + (9 \tan \theta)^2)^2} \]
Simplified, it becomes:
\[ \int \frac{(9^4 \tan^2 \theta \sec^2 \theta \, d\theta)}{(9^4 (1 + \tan^2 \theta)^2)} \]
Further simplifying the expression, we get:
\[ \int \frac{\tan^2 \theta \sec^2 \theta \, d\theta}{(\sec^2 \theta)^2} \]
Note how trigonometric identities and substitutions help in transforming the integral into a more manageable form. This entire process often simplifies the integral to a standard form that can be easily evaluated.
---
Notice that the image contains hand-written steps that include trigonometric substitution and algebraic manipulation to simplify the given integral. The integral used is common in calculus courses dealing with integration techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66c7b497-dac7-4855-b923-2e60bbc73063%2F5d11deaf-f374-443c-800f-7049a16942cc%2Fyerdidp.jpeg&w=3840&q=75)
Transcribed Image Text:### Transcription for Educational Website
#### Calculus Question: Trigonometric Substitution in Integration
We start with the integral:
\[ \int \frac{9x^2}{(81 + x^2)^2} \, dx \]
Using the substitution \[ x = 9 \, \tan \theta \], we find:
\[ dx = 9 \, \sec^2 \theta \, d\theta \]
This transforms the integral into:
\[ \int \frac{9 (9 \tan \theta)^2 \cdot 9 \sec^2 \theta \, d\theta}{(81 + (9 \tan \theta)^2)^2} \]
Simplified, it becomes:
\[ \int \frac{(9^4 \tan^2 \theta \sec^2 \theta \, d\theta)}{(9^4 (1 + \tan^2 \theta)^2)} \]
Further simplifying the expression, we get:
\[ \int \frac{\tan^2 \theta \sec^2 \theta \, d\theta}{(\sec^2 \theta)^2} \]
Note how trigonometric identities and substitutions help in transforming the integral into a more manageable form. This entire process often simplifies the integral to a standard form that can be easily evaluated.
---
Notice that the image contains hand-written steps that include trigonometric substitution and algebraic manipulation to simplify the given integral. The integral used is common in calculus courses dealing with integration techniques.
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