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c. (1+3lnz)dx 1. Find the followingu= a. S(-4t + 1)³dt du= %3D b. fre¯*dr = Se™d U=1+ (1+3ln x)' с. du=

c. (1+3lnz)dx 1. Find the followingu= a. S(-4t + 1)³dt du= %3D b. fre¯*dr = Se™d U=1+ (1+3ln x)' с. du=

Calculus: Early Transcendentals
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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**1. Find the following:** a. \(\int (-4t + 1)^3 dt\) b. \(\int r e^{-r^2} dr = \int e^{-r^2} du\) c. \(\int \frac{(1 + 3 \ln x)^2}{x} dx\) d. \(\int_{-1}^{0} y (2y^2 - 3)^5 dy\) --- **Explanation of Equations:** - **Equation a** involves integrating a polynomial expression. - **Equation b** features an integral involving an exponential function with a Gaussian term. The transformation shows a substitution method is being applied, where \(u\) is a function of \(r\), and \(du\) represents the differential change. - **Equation c** is a rational function inside an integral, indicating a substitution technique might be applicable for integration. - **Equation d** presents a definite integral with limits from \(-1\) to \(0\), involving a polynomial raised to a power. **Additional Information:** - The blue annotations suggest standard integration techniques such as substitution. These would typically be found in a calculus context focusing on integration methods. - Notably, items such as \(u=\) and \(du=\) in annotations indicate specific substitutions for solving the integrals, though explicit expressions for these are not given.
Transcribed Image Text:**1. Find the following:** a. \(\int (-4t + 1)^3 dt\) b. \(\int r e^{-r^2} dr = \int e^{-r^2} du\) c. \(\int \frac{(1 + 3 \ln x)^2}{x} dx\) d. \(\int_{-1}^{0} y (2y^2 - 3)^5 dy\) --- **Explanation of Equations:** - **Equation a** involves integrating a polynomial expression. - **Equation b** features an integral involving an exponential function with a Gaussian term. The transformation shows a substitution method is being applied, where \(u\) is a function of \(r\), and \(du\) represents the differential change. - **Equation c** is a rational function inside an integral, indicating a substitution technique might be applicable for integration. - **Equation d** presents a definite integral with limits from \(-1\) to \(0\), involving a polynomial raised to a power. **Additional Information:** - The blue annotations suggest standard integration techniques such as substitution. These would typically be found in a calculus context focusing on integration methods. - Notably, items such as \(u=\) and \(du=\) in annotations indicate specific substitutions for solving the integrals, though explicit expressions for these are not given.
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