4 Evaluate the indefinite integral by using the substitution u = y² + 4y² + 2 to reduce the integral to standard form. [12(y² + 4y² + 2)² (y² + 2y) dy [12 (y² + 4y² + 2)² (y² + 2y) dy =
4 Evaluate the indefinite integral by using the substitution u = y² + 4y² + 2 to reduce the integral to standard form. [12(y² + 4y² + 2)² (y² + 2y) dy [12 (y² + 4y² + 2)² (y² + 2y) dy =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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![**Title: Evaluating Indefinite Integrals Using Substitution**
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In this example, we will evaluate the indefinite integral by using the substitution \( u = y^4 + 4y^2 + 2 \) to reduce the integral to standard form.
The integral to be evaluated is:
\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy \]
To proceed, we use the given substitution \( u = y^4 + 4y^2 + 2 \).
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\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy = 12 \int (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy \]
Please input your solution for the given integral transformation.
\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy = \]
---
This section helps students understand how using substitution can simplify complex integrals, reducing them to a more manageable form. Students are encouraged to attempt the integral using the substitution method and verify their results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c5e517d-57db-4b65-baa5-8a292d97b7f5%2F8c782022-28ef-4eee-abf6-170a24a4d633%2Fi3l3g8_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating Indefinite Integrals Using Substitution**
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In this example, we will evaluate the indefinite integral by using the substitution \( u = y^4 + 4y^2 + 2 \) to reduce the integral to standard form.
The integral to be evaluated is:
\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy \]
To proceed, we use the given substitution \( u = y^4 + 4y^2 + 2 \).
---
\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy = 12 \int (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy \]
Please input your solution for the given integral transformation.
\[ \int 12 (y^4 + 4y^2 + 2)^2 (y^3 + 2y) \, dy = \]
---
This section helps students understand how using substitution can simplify complex integrals, reducing them to a more manageable form. Students are encouraged to attempt the integral using the substitution method and verify their results.
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