### Integral ExercisesOn 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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Answered: J(2t-3)-e-t²,3t i. 2t – dt 3 2vlnx dx .…

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

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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Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

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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} \

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