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.

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.