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The image shows a table with six integrals, and for each integral, you need to determine suitable substitutions of the form \( u = g(x) \) and \( du = g'(x) dx \). Here's the analysis and suggested substitutions for each integral:

1. **\(\int (6x^2 + 1)^3 (6x) \, dx\)**
   - Suggested substitution:
     - \( u = 6x^2 + 1 \)
     - \( du = 12x \, dx \)

2. **\(\int \frac{x^2}{\sqrt{x^2 + 1}} \, dx\)**
   - Suggested substitution:
     - \( u = x^2 + 1 \)
     - \( du = 2x \, dx \)

3. **\(\int \frac{x}{x^2 + 1} \, dx\)**
   - Suggested substitution:
     - \( u = x^2 + 1 \)
     - \( du = 2x \, dx \)

4. **\(\int \sec 2x \tan 2x \, dx\)**
   - Suggested substitution:
     - \( u = 2x \)
     - \( du = 2 \, dx \)

5. **\(\int \tan^2 x \sec^2 x \, dx\)**
   - Suggested substitution:
     - \( u = \tan x \)
     - \( du = \sec^2 x \, dx \)

6. **\(\int \frac{\cos x}{\sin^2 x} \, dx\)**
   - Suggested substitution:
     - \( u = \sin x \)
     - \( du = \cos x \, dx \)

Would you like more details on any of these substitutions or how they are applied in integration?

Here are some related questions for further exploration:

1. How do you decide the substitution \( u \) in an integral problem?
2. What are common substitution strategies for trigonometric integrals?
3. How can substitution simplify the process of solving integrals?
4. Can substitution be used in definite integrals, and how does it affect the limits?
5. What are some pitfalls to avoid when using substitution in integrals?

**Tip:** Always verify the differential \( du \) matches the terms in the original integral to ensure a correct substitution.

Determine suitable substitutions of the form u = g(x) and du = g'(x) dx for the given integrals.

Learn how to find the appropriate substitutions for solving integrals using the substitution method. This problem involves six integrals and provides step-by-step solutions for determining u and du, focusing on calculus concepts like the substitution method and integration techniques. Suitable for advanced high school or undergraduate students.

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