Can someone help me with this question pls and write neatly so I can understand please ---### Integration Problem with Detailed Solution#### Problem:Evaluate the integral. (Remember to use absolute values where appropriate. Use \( C \) for the constant of integration.)\[\int \frac{7}{x^2 - 25} \, dx\]#### Solution:\[ \frac{7}{7\sqrt{5}} \left[ \ln|x + 5| - \frac{1}{2} \ln(x^2 - 5x + 25) - \sqrt{3} \, \arctan \left( \frac{2}{5 \sqrt{3}}\left( x + \frac{5}{2} \right) \right)\right] + C \]---### Explanation:The integral provided in the problem consists of a fraction with a polynomial in the denominator. The integration process might involve techniques such as polynomial division, partial fraction decomposition, or trigonometric substitution to simplify and solve the integral.- **Natural Logarithms**: Observed terms like \(\ln|x + 5|\) and \(\ln(x^2 - 5x + 25)\) indicate the use of natural logarithms in the solution.- **Arc Tangent Function**: The term involving \(\arctan\) suggests a trigonometric substitution was used during the integration process.- **Absolute Values**: Note the use of absolute values for the logarithmic terms to ensure the function's domain remains valid.### Constant of Integration:Don't forget to add the constant of integration, \( C \), which is a crucial part of any indefinite integral.---This detailed breakdown would help students understand the steps taken to solve the integral and the rationale behind each part of the solution.

Answered: Image /qna-images/answer/65392b3c-7c56-48c1-b152-c044a9bf84a5.jpg

Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx X²³-125 2 75 - In(x + 5) – ½ln(n² + 5x + 25 ) – ✓arctan + Bx + 28 ) - và arctan Bảo (x+ + 5√3

Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) dx X²³-125 2 75 - In(x + 5) – ½ln(n² + 5x + 25 ) – ✓arctan + Bx + 28 ) - và arctan Bảo (x+ + 5√3

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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Can someone help me with this question pls and write neatly so I can understand please

---

### Integration Problem with Detailed Solution

#### Problem:
Evaluate the integral. (Remember to use absolute values where appropriate. Use \( C \) for the constant of integration.)

\[
\int \frac{7}{x^2 - 25} \, dx
\]

#### Solution:
\[
\frac{7}{7\sqrt{5}} \left[ \ln|x + 5| - \frac{1}{2} \ln(x^2 - 5x + 25) - \sqrt{3} \, \arctan \left( \frac{2}{5 \sqrt{3}}\left( x + \frac{5}{2} \right) \right)\right] + C
\]

---

### Explanation:
The integral provided in the problem consists of a fraction with a polynomial in the denominator. The integration process might involve techniques such as polynomial division, partial fraction decomposition, or trigonometric substitution to simplify and solve the integral.

- **Natural Logarithms**: Observed terms like \(\ln|x + 5|\) and \(\ln(x^2 - 5x + 25)\) indicate the use of natural logarithms in the solution.
- **Arc Tangent Function**: The term involving \(\arctan\) suggests a trigonometric substitution was used during the integration process.
- **Absolute Values**: Note the use of absolute values for the logarithmic terms to ensure the function's domain remains valid.

### Constant of Integration:
Don't forget to add the constant of integration, \( C \), which is a crucial part of any indefinite integral.

---

This detailed breakdown would help students understand the steps taken to solve the integral and the rationale behind each part of the solution.

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