How can I get rid of the highlighted x in this u-substitution problem? ## Integration by Substitution ExampleThis example demonstrates how to use substitution for the integration of $ \int \sec^3{x} \tan{x} \, dx $.### Step-by-Step Solution:1. Begin with the integral: \[ \int \sec^3{x} \tan{x} \, dx \]2. Introduce a substitution: Let $ u = \tan{x} $ which implies that $ du = \sec^2{x} \, dx $. Rearranging for $ dx $, we get: \[ dx = \frac{1}{\sec^2{x}} \, du \]3. Substitute $ u $ and $ dx $ into the integral: Since $ \sec^3{x} = \sec{x} \cdot \sec^2{x} $, the integral becomes: \[ \int \frac{\sec^3{x} \cdot u}{\sec^2{x}} \, du \]4. Simplify the expression: \[ \int \sec{x} \cdot u \, du \] At this point, there is a remaining $ \sec{x} $ term, despite substituting $ dx $.5. Identifying the problem: There is a highlighted text in the image indicating: *How to get rid of this $ x $?* To handle the extra $\sec{x}$ term in $ \sec{x} \cdot u \, du $, note that $\sec{x} = \sqrt{u^2 + 1}$ because $\sec^2{x} = 1 + \tan^2{x}$.### SummaryTo complete the integral, substitute $\sec{x}$ using the relation to $ u $:\[\sec{x} = \sqrt{1 + u^2}\]Transforming the integral fully in terms of $ u $:\[\int \sqrt{1 + u^2} \, u \, du\]The final result requires further simplification or a different technique to integrate the expression completely.

Answered: Image /qna-images/answer/1cb226ee-3f7c-49d9-bf8a-40cf5f54b102.jpg

Answered: sec 3 x tan x (sec ³x sec³x secrx S see…

sec 3 x tan x (sec ³x sec³x secrx S see of secxu S u ли secx u du dx let a = tan x du = sec² x dx dx= 1 Sec² x du -How to get rid of this x?

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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