Consider the indefinite integral ¹ / = (x5 - 2)5 This can be transformed into a basic integral by letting U = du = and du dx dx: Performing the substitution yields the integral
Consider the indefinite integral ¹ / = (x5 - 2)5 This can be transformed into a basic integral by letting U = du = and du dx dx: Performing the substitution yields the integral
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 and Explanation for Educational Website**
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**Topic:** Substitution in Calculus - Indefinite Integrals
Consider the indefinite integral:
\[
\int \frac{x^4}{(x^5 - 2)^5} \, dx
\]
**Transformation through Substitution:**
This integral can be simplified by using substitution. To do this, we let:
\[
u = \boxed{x^5 - 2}
\]
This means that the differential \( du \) is:
\[
du = \boxed{5x^4} \, dx
\]
**Performing the Substitution:**
With the substitution in place, the integral becomes:
\[
\int \boxed{\frac{1}{5u^5}} \, du
\]
**Explanation:**
The substitution method involves changing variables to simplify the integration process. In this example, the goal is to transform the complex expression into a simpler form that is easier to integrate. By identifying a part of the expression within the integral as \( u \), and consequently computing \( du \), the integral becomes manageable. Performing the integration on the simpler expression allows for finding the indefinite integral more efficiently. This is a common approach in calculus to handle integrals involving polynomial and rational expressions.
---
For more detailed examples and explanations, refer to the [Calculus Section] of our educational resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0123591-fed7-4568-91c4-d5c32b5e453b%2Ff3b6a373-33a8-4740-bd84-7494da57ccba%2Fizli81_processed.png&w=3840&q=75)
Transcribed Image Text:**Transcription and Explanation for Educational Website**
---
**Topic:** Substitution in Calculus - Indefinite Integrals
Consider the indefinite integral:
\[
\int \frac{x^4}{(x^5 - 2)^5} \, dx
\]
**Transformation through Substitution:**
This integral can be simplified by using substitution. To do this, we let:
\[
u = \boxed{x^5 - 2}
\]
This means that the differential \( du \) is:
\[
du = \boxed{5x^4} \, dx
\]
**Performing the Substitution:**
With the substitution in place, the integral becomes:
\[
\int \boxed{\frac{1}{5u^5}} \, du
\]
**Explanation:**
The substitution method involves changing variables to simplify the integration process. In this example, the goal is to transform the complex expression into a simpler form that is easier to integrate. By identifying a part of the expression within the integral as \( u \), and consequently computing \( du \), the integral becomes manageable. Performing the integration on the simpler expression allows for finding the indefinite integral more efficiently. This is a common approach in calculus to handle integrals involving polynomial and rational expressions.
---
For more detailed examples and explanations, refer to the [Calculus Section] of our educational resources.
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