Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 54CR
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Question
Evaluate the
![Certainly! Below is the transcription of the mathematical expression provided in the image:
---
### Integral of the Function
We need to evaluate the integral:
\[ \int 4x \sec^2(2x) \, dx \]
To solve this integral, a suitable substitution method can be employed to simplify the expression for easier integration.
### Steps to Solve
1. **Substitution**:
Define \( u = 2x \). Then \( du = 2dx \) or \( dx = \frac{1}{2} du \).
2. **Rewrite the integral**:
Substitute \( u \) and \( du \) into the integral:
\[ \int 4x \sec^2(2x) \, dx = \int 4 \cdot \frac{u}{2} \sec^2(u) \cdot \frac{1}{2} \, du \]
Simplify the integrand:
\[ = \int 2u \sec^2(u) \cdot \frac{1}{2} \, du = \int u \sec^2(u) \, du \]
3. **Integration by parts (if needed)**:
At this point, if the direct integration seems complex, consider applying integration by parts.
This process continues by applying the necessary integration techniques to solve for the antiderivative. This specific example might bring into context several key methods in calculus such as substitution and integration by parts.
For more detailed step-by-step solutions and explanations, you can explore additional resources or consult with a math tutor.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F790e387a-579c-4b02-b6cb-cbab761f0d7e%2F3d2ca3cd-b87e-4f1c-9b12-8a40430a8924%2Fuise5j.png&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcription of the mathematical expression provided in the image:
---
### Integral of the Function
We need to evaluate the integral:
\[ \int 4x \sec^2(2x) \, dx \]
To solve this integral, a suitable substitution method can be employed to simplify the expression for easier integration.
### Steps to Solve
1. **Substitution**:
Define \( u = 2x \). Then \( du = 2dx \) or \( dx = \frac{1}{2} du \).
2. **Rewrite the integral**:
Substitute \( u \) and \( du \) into the integral:
\[ \int 4x \sec^2(2x) \, dx = \int 4 \cdot \frac{u}{2} \sec^2(u) \cdot \frac{1}{2} \, du \]
Simplify the integrand:
\[ = \int 2u \sec^2(u) \cdot \frac{1}{2} \, du = \int u \sec^2(u) \, du \]
3. **Integration by parts (if needed)**:
At this point, if the direct integration seems complex, consider applying integration by parts.
This process continues by applying the necessary integration techniques to solve for the antiderivative. This specific example might bring into context several key methods in calculus such as substitution and integration by parts.
For more detailed step-by-step solutions and explanations, you can explore additional resources or consult with a math tutor.
---
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