Consider the integral / 3x* (25 + 1) dz. In the following, we will evaluate the integral using two methods. A. First, rewrite the integral by multiplying out the integrand: | 32* (z* + 1) dz =| 3x°9+3x*4 dz Then evaluate the resulting integral term-by-term: | 32 (2* + 1) dz = B. Next, rewrite the integral using the substitution w = 25 +1: | 3z* (z* + 1) dz =|| dw Evaluate this integral (and back-substitute for w) to find the value of the original integral: | 32* (2* + 1) dr =

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...
Question
Consider the integral
3x* (x° + 1) dr. In the following, we will evaluate the integral using two methods.
A. First, rewrite the integral by multiplying out the integrand:
/ 32* (* + 1) dz = /
3x4
3x^9+3x^4
dr
Then evaluate the resulting integral term-by-term:
| 32 (25 + 1) da =
3x4
B. Next, rewrite the integral using the substitution w = x +1:
25
3z* (r + 1) dx =||
dw
Evaluate this integral (and back-substitute for w) to find the value of the original integral:
3x* (x + 1) dx
C. How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.)
(answer from B)-(answer from A) =
Are both of the answers correct? (Be sure you can explain why they are!)
Transcribed Image Text:Consider the integral 3x* (x° + 1) dr. In the following, we will evaluate the integral using two methods. A. First, rewrite the integral by multiplying out the integrand: / 32* (* + 1) dz = / 3x4 3x^9+3x^4 dr Then evaluate the resulting integral term-by-term: | 32 (25 + 1) da = 3x4 B. Next, rewrite the integral using the substitution w = x +1: 25 3z* (r + 1) dx =|| dw Evaluate this integral (and back-substitute for w) to find the value of the original integral: 3x* (x + 1) dx C. How are your expressions from parts (A) and (B) different? What is the difference between the two? (Ignore the constant of integration.) (answer from B)-(answer from A) = Are both of the answers correct? (Be sure you can explain why they are!)
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