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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![**Title: Evaluating Integrals Using U-Substitution**
**Instructions:**
Use u-substitution to evaluate the integral.
**Integral Expression:**
\[ \int \left(x^{\frac{3}{2}} + 5\right)^6 \sqrt{x} \, dx \]
**Solution:**
Evaluate the integral:
\[ \int \left(x^{\frac{3}{2}} + 5\right)^6 \sqrt{x} \, dx = \boxed{\hspace{1cm}} \]
**Explanation:**
This exercise requires the application of u-substitution, a method used in calculus to simplify the process of integrating composite functions. To tackle this problem, identify a substitution that will simplify the integrand, generally by letting \( u = x^{\frac{3}{2}} + 5 \). Then, compute the derivative \( du \) and substitute back into the integral in terms of \( u \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3f88845-b007-4fd1-915c-0698883559f2%2F3bf270fd-827e-4552-a8ba-624e65685c7e%2Frhkgncj_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating Integrals Using U-Substitution**
**Instructions:**
Use u-substitution to evaluate the integral.
**Integral Expression:**
\[ \int \left(x^{\frac{3}{2}} + 5\right)^6 \sqrt{x} \, dx \]
**Solution:**
Evaluate the integral:
\[ \int \left(x^{\frac{3}{2}} + 5\right)^6 \sqrt{x} \, dx = \boxed{\hspace{1cm}} \]
**Explanation:**
This exercise requires the application of u-substitution, a method used in calculus to simplify the process of integrating composite functions. To tackle this problem, identify a substitution that will simplify the integrand, generally by letting \( u = x^{\frac{3}{2}} + 5 \). Then, compute the derivative \( du \) and substitute back into the integral in terms of \( u \).
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