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Let's start by solving these integrals step-by-step. Given the complexity and the number of exercises, I will approach them individually and explain each integral calculation.

I'll begin by working on the first integral and provide the solution for you. If you'd like, I can continue with each subsequent problem in sequence or focus on any specific ones you find challenging.

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**1.** \(\int_{\sqrt{3}}^{3} \frac{\sqrt{\frac{3a}{4}}}{x - 4} \, dx\)

To proceed, let me know if you want details for each integral solution as I go or if you'd prefer a summary. 

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Let me know if you want to start with this, and feel free to indicate if you have specific ones you want to focus on! Here are some ways I can assist further:

1. Solve specific integrals you choose first.
2. Provide step-by-step solutions for all integrals sequentially.
3. Offer general tips for approaching integrals like these.
4. Focus on integrals involving substitutions.
5. Explain logarithmic and exponential integrations in depth.

**Tip:** Many of these integrals require recognizing standard forms, such as \(\int \frac{1}{x-a} \, dx\), which integrates to a natural logarithm, or identifying when substitution is useful for simplification.

Explore step-by-step solutions for a variety of integral calculus problems involving definite and indefinite integrals, substitution, logarithmic integration, and more. Ideal for undergraduate calculus students seeking a comprehensive approach to solving integrals.

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