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J(2t-3)-e-t²,3t i. 2t – dt 3 2vlnx dx . ii. X 2 iii. (2y-3]12 kdy.

J(2t-3)-e-t²,3t i. 2t – dt 3 2vlnx dx . ii. X 2 iii. (2y-3]12 kdy.

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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### Integral Exercises On this page, you will find various integral problems designed to test and enhance your understanding of integral calculus. Below are three integrals each with unique challenges: 1. **Integral \( \int (2t - 3) \cdot e^{-t^2 + 3t} \, dt \)** This integral involves a combination of a linear polynomial and an exponential function with a quadratic polynomial in the exponent. 2. **Integral \( \int_{2}^{3} \left( \frac{2 \sqrt{\ln x}}{x} \right) \, dx \)** This is a definite integral, with limits from 2 to 3, involving a function \( \frac{2 \sqrt{\ln x}}{x} \). Note how the natural logarithm is present under the square root, which adds a layer of complexity. 3. **Integral \( \int \left( \frac{y}{(2y - 3)^{1/2}} \right) \, dy \)** In this integral, the numerator is a linear function of \( y \), while the denominator is the square root of a linear polynomial. ### Explanation of Concepts: 1. **Polynomial and Exponential Function**: - The first integral pairs a polynomial and an exponential function. Such integrals often require integration by parts or a substitution method to simplify the process. 2. **Definite Integral with Logarithmic and Radical Functions**: - The second integral requires evaluating a specific interval, adding complexity because of the natural logarithm and the square root operation. Techniques such as substitution (e.g., letting \( u = \ln x \)) are often useful. 3. **Integration Involving Radicals**: - The third integral involves a rational expression with a square root denominator. Often, substitution simplifies this to a familiar form, enabling easier integration. ### Visual Explanation: While integral problems primarily focus on analytical solutions, visual aids such as graphs can provide intuitive understanding: 1. **Graph of the function \( (2t - 3) \cdot e^{-t^2 + 3t} \)**: - Plotting helps visualize the behavior of the function, particularly where it crosses the t-axis or asymptotically approaches limits. 2. **Graph of the function \( \frac{2 \sqrt{\ln x}}{x} \
Transcribed Image Text:### Integral Exercises On this page, you will find various integral problems designed to test and enhance your understanding of integral calculus. Below are three integrals each with unique challenges: 1. **Integral \( \int (2t - 3) \cdot e^{-t^2 + 3t} \, dt \)** This integral involves a combination of a linear polynomial and an exponential function with a quadratic polynomial in the exponent. 2. **Integral \( \int_{2}^{3} \left( \frac{2 \sqrt{\ln x}}{x} \right) \, dx \)** This is a definite integral, with limits from 2 to 3, involving a function \( \frac{2 \sqrt{\ln x}}{x} \). Note how the natural logarithm is present under the square root, which adds a layer of complexity. 3. **Integral \( \int \left( \frac{y}{(2y - 3)^{1/2}} \right) \, dy \)** In this integral, the numerator is a linear function of \( y \), while the denominator is the square root of a linear polynomial. ### Explanation of Concepts: 1. **Polynomial and Exponential Function**: - The first integral pairs a polynomial and an exponential function. Such integrals often require integration by parts or a substitution method to simplify the process. 2. **Definite Integral with Logarithmic and Radical Functions**: - The second integral requires evaluating a specific interval, adding complexity because of the natural logarithm and the square root operation. Techniques such as substitution (e.g., letting \( u = \ln x \)) are often useful. 3. **Integration Involving Radicals**: - The third integral involves a rational expression with a square root denominator. Often, substitution simplifies this to a familiar form, enabling easier integration. ### Visual Explanation: While integral problems primarily focus on analytical solutions, visual aids such as graphs can provide intuitive understanding: 1. **Graph of the function \( (2t - 3) \cdot e^{-t^2 + 3t} \)**: - Plotting helps visualize the behavior of the function, particularly where it crosses the t-axis or asymptotically approaches limits. 2. **Graph of the function \( \frac{2 \sqrt{\ln x}}{x} \
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