Let's solve the given integral:

Problem

Evaluate: $\int \sec \left(x + \frac{\pi}{4}\right) dx$

Solution:

The integral of $\sec u$ is a standard integral formula: $\int \sec u \, du = \ln |\sec u + \tan u| + C$

Here:

• $u = x + \frac{\pi}{4}$
• $du = dx$

Substitute $u = x + \frac{\pi}{4}$ into the integral: $\int \sec \left(x + \frac{\pi}{4}\right) dx = \ln \left| \sec \left(x + \frac{\pi}{4}\right) + \tan \left(x + \frac{\pi}{4}\right) \right| + C$

Final Answer:

$\int \sec \left(x + \frac{\pi}{4}\right) dx = \ln \left| \sec \left(x + \frac{\pi}{4}\right) + \tan \left(x + \frac{\pi}{4}\right) \right| + C$

Would you like a detailed explanation of the steps? Let me know!

Here are some related questions you might find helpful:

1. What are some common trigonometric integral formulas?
2. How can we integrate functions involving tangent or cotangent?
3. What are the properties of $\sec(x)$ and $\tan(x)$?
4. How is substitution used in integration problems?
5. Can we verify integration results by differentiating the solution?

Tip: Always keep a list of standard integral formulas for trigonometric functions—it saves time during exams!