Question 2 of the attachment. If you could show the work that would be helpful. ### Calculus Integration ProblemsHere are a set of integral problems that can be used for practice in an educational setting. They vary in difficulty and cover a range of integral types.1. $\int (2 - x^2)^4 \; dx$2. $\int \frac{x \; dx}{9 + x^2}$3. $\int \frac{x^2 - 5x + 6 \; dx}{x^3 - 2x^2 + x}$4. $\int x e^{\frac{3}{5} x} \; dx$5. $\int x^3 \ln x \; dx$6. $\int \cos 5x \sin 9x \; dx$7. $\int \tan^4 \left( \frac{5}{x} \right) \; dx$### Explanations and Graphs (if applicable)Each integral challenges the student to apply different integration techniques including polynomial integration, integration by parts, and trigonometric integral methods.1. **Polynomial Integration**: - The first integral, $\int (2 - x^2)^4 \; dx$, requires binomial expansion or a substitution method to simplify before integrating.2. **Integration Using Substitution**: - The second integral, $\int \frac{x \; dx}{9 + x^2}$, may benefit from a simple substitution to simplify the ratio into a more straightforward integral.3. **Partial Fraction Decomposition**: - In the third integral, $\int \frac{x^2 - 5x + 6 \; dx}{x^3 - 2x^2 + x}$, partial fraction decomposition can be utilized to break down the complex fraction before integrating term-by-term.4. **Integration by Parts**: - For the fourth integral, $\int x e^{\frac{3}{5} x} \; dx$, integration by parts would be an effective method to solve it.5. **Logarithmic Integration**: - The fifth integral, $\int x^3 \ln x \; dx$, is another case where integration by parts will be essential due to the presence of the logarithmic function.6. **Trigonometric Integral**: - The sixth integral, \(\int \cos

Given integral is We will use following substitution to evaluate the integral

Answered: 2-[xdx 9tx2

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Calculus

2-[xdx 9tx2

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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Question 2 of the attachment. If you could show the work that would be helpful.

### Calculus Integration Problems

Here are a set of integral problems that can be used for practice in an educational setting. They vary in difficulty and cover a range of integral types.

1. $\int (2 - x^2)^4 \; dx$
2. $\int \frac{x \; dx}{9 + x^2}$
3. $\int \frac{x^2 - 5x + 6 \; dx}{x^3 - 2x^2 + x}$
4. $\int x e^{\frac{3}{5} x} \; dx$
5. $\int x^3 \ln x \; dx$
6. $\int \cos 5x \sin 9x \; dx$
7. $\int \tan^4 \left( \frac{5}{x} \right) \; dx$

### Explanations and Graphs (if applicable)

Each integral challenges the student to apply different integration techniques including polynomial integration, integration by parts, and trigonometric integral methods.

1. **Polynomial Integration**:
- The first integral, $\int (2 - x^2)^4 \; dx$, requires binomial expansion or a substitution method to simplify before integrating.

2. **Integration Using Substitution**:
- The second integral, $\int \frac{x \; dx}{9 + x^2}$, may benefit from a simple substitution to simplify the ratio into a more straightforward integral.

3. **Partial Fraction Decomposition**:
- In the third integral, $\int \frac{x^2 - 5x + 6 \; dx}{x^3 - 2x^2 + x}$, partial fraction decomposition can be utilized to break down the complex fraction before integrating term-by-term.

4. **Integration by Parts**:
- For the fourth integral, $\int x e^{\frac{3}{5} x} \; dx$, integration by parts would be an effective method to solve it.

5. **Logarithmic Integration**:
- The fifth integral, $\int x^3 \ln x \; dx$, is another case where integration by parts will be essential due to the presence of the logarithmic function.

6. **Trigonometric Integral**:
- The sixth integral, \(\int \cos

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