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Let's go through each problem step-by-step to provide solutions and explanations for the integrals in your image.

### Problem 11
**Evaluate** \( \int \frac{1}{1 + \sec x} \, dx \).

Rewrite \( 1 + \sec x \) in terms of trigonometric identities and simplify the integrand. Then solve the integral.

### Problem 12
**Evaluate** \( \int \frac{1 + \cos x}{1 - \cos x} \, dx \).

To solve this integral, use trigonometric identities, such as converting \( \cos x \) in terms of \( \sin^2 \left(\frac{x}{2}\right) \) to simplify.

### Problem 13
**Evaluate** \( \int \frac{1 + \sin 2x}{x + \sin^2 x} \, dx \).

For this integral, consider using a substitution to simplify the expression. Since \( \sin 2x = 2 \sin x \cos x \), use this identity and try substitutions that simplify \( \sin^2 x \).

### Problem 14
**Evaluate** \( \int \cos 4x \cos x \, dx \).

Use the product-to-sum identities to rewrite \( \cos 4x \cos x \) as a sum of cosines, which makes it easier to integrate term by term.

### Problem 15
**Evaluate** \( \int x \sec^8 x \, dx \).

Here, you might consider using integration by parts or a trigonometric substitution to deal with the \( x \sec^8 x \) term.

### Problem 16
**Evaluate** \( \int x \cos^2 x \, dx \).

Rewrite \( \cos^2 x \) in terms of \( \cos 2x \) using the identity \( \cos^2 x = \frac{1 + \cos 2x}{2} \), then apply integration by parts.

### Problem 17
**Evaluate** \( \int \frac{\cos x - \cos 2x}{1 - \cos x} \, dx \).

Simplify the integrand by expressing \( \cos 2x \) and \( \cos x \) using trigonometric identities, and then try to separate the terms to simplify the integral.

### Problem 18
**Evaluate** \( \int x^2 \cos x \, dx \).

Use integration by parts, letting \( u = x^2 \) and \( dv = \cos x \, dx \). This will involve applying integration by parts multiple times.

### Problem 19
**Evaluate** \( \int \frac{2x + 1}{(x + 1)(x - 2)} \, dx \).

Perform partial fraction decomposition on \( \frac{2x + 1}{(x + 1)(x - 2)} \) and then integrate each term separately.

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Would you like detailed solutions for each of these integrals or specific ones? Let me know which ones to focus on!

### Here are 5 related questions for further practice:
1. How would you apply trigonometric identities to simplify integrals with products of sine and cosine?
2. What are the steps to perform partial fraction decomposition on a rational function?
3. How do integration by parts and substitution differ, and when should each be used?
4. Can you solve an integral involving \( \sec^n x \) by rewriting it in terms of \( \tan x \)?
5. What is the general approach for evaluating integrals with powers of trigonometric functions?

### Tip:
When faced with trigonometric integrals, remember to check for identities or substitutions that can simplify the integrand before attempting more complex techniques like integration by parts.

Evaluate the following integrals:
1. \( \int \frac{1}{1 + \sec x} \, dx \)
2. \( \int \frac{1 + \cos x}{1 - \cos x} \, dx \)
3. \( \int \frac{1 + \sin 2x}{x + \sin^2 x} \, dx \)
4. \( \int \cos 4x \cos x \, dx \)
5. \( \int x \sec^8 x \, dx \)
6. \( \int x \cos^2 x \, dx \)
7. \( \int \frac{\cos x - \cos 2x}{1 - \cos x} \, dx \)
8. \( \int x^2 \cos x \, dx \)
9. \( \int \frac{2x + 1}{(x + 1)(x - 2)} \, dx \).

Solve complex integrals including trigonometric and rational functions. This detailed guide covers integration techniques like partial fraction decomposition, trigonometric identities, and integration by parts, perfect for undergraduate calculus students.

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