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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Please answer the questions with steps and correct answer.
![**Problem Statement**
Use an appropriate substitution to evaluate the integral:
\[
\int x \sqrt{5x + 2} \, dx.
\]
**Solution Approach**
To solve the integral, identify a substitution that simplifies the expression under the square root. Set \( u = 5x + 2 \), which implies \( du = 5 \, dx \) or \( dx = \frac{du}{5} \).
Next, express \( x \) in terms of \( u \):
\[ 5x + 2 = u \implies x = \frac{u - 2}{5}. \]
Now, substitute back into the integral:
\[
\int x \sqrt{5x + 2} \, dx = \int \frac{u - 2}{5} \sqrt{u} \cdot \frac{du}{5}.
\]
This substitution transforms the integral into a form that is easier to evaluate. Simplify and integrate to find the solution.
**Further Explanation**
Such problems help in understanding the application of the substitution method, a fundamental technique in calculus used to simplify and compute complex integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ba37d91-3081-42ea-952f-3d2c5579a63d%2Fe0a74681-c24e-4288-b435-aac3609844ba%2Fsutiemwu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Use an appropriate substitution to evaluate the integral:
\[
\int x \sqrt{5x + 2} \, dx.
\]
**Solution Approach**
To solve the integral, identify a substitution that simplifies the expression under the square root. Set \( u = 5x + 2 \), which implies \( du = 5 \, dx \) or \( dx = \frac{du}{5} \).
Next, express \( x \) in terms of \( u \):
\[ 5x + 2 = u \implies x = \frac{u - 2}{5}. \]
Now, substitute back into the integral:
\[
\int x \sqrt{5x + 2} \, dx = \int \frac{u - 2}{5} \sqrt{u} \cdot \frac{du}{5}.
\]
This substitution transforms the integral into a form that is easier to evaluate. Simplify and integrate to find the solution.
**Further Explanation**
Such problems help in understanding the application of the substitution method, a fundamental technique in calculus used to simplify and compute complex integrals.
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