To solve the integral $\int \sec(x) \, dx$, we use a standard technique involving multiplying by a clever form of 1, which is:

$\int \sec(x) \, dx = \int \frac{\sec(x)(\sec(x) + \tan(x))}{\sec(x) + \tan(x)} \, dx$

Now, let’s proceed with the steps:

Step 1: Multiply by the expression

$\int \frac{\sec(x)(\sec(x) + \tan(x))}{\sec(x) + \tan(x)} \, dx = \int \frac{d(\sec(x) + \tan(x))}{\sec(x) + \tan(x)}$ This works because $d(\sec(x) + \tan(x)) = \sec(x)\tan(x) + \sec^2(x) \, dx$, which simplifies the problem.

Step 2: Substitute

Let: $u = \sec(x) + \tan(x)$ Then: $du = (\sec(x)\tan(x) + \sec^2(x)) \, dx$ Thus, the integral becomes: $\int \frac{du}{u} = \ln|u| + C$

Step 3: Substitute back

Substitute $u = \sec(x) + \tan(x)$ back into the result: $\int \sec(x) \, dx = \ln|\sec(x) + \tan(x)| + C$

So the solution is: $\boxed{\ln|\sec(x) + \tan(x)| + C}$

Would you like more details or have any further questions?

Here are 5 related questions to explore:

1. How would you compute $\int \sec^2(x) \, dx$?
2. What is the derivative of $\ln|\sec(x) + \tan(x)|$?
3. Can you compute $\int \sec^3(x) \, dx$?
4. What is the integral of $\int \csc(x) \, dx$?
5. How does the substitution method help simplify integrals involving trigonometric functions?

Tip: When dealing with integrals of trigonometric functions, try to recall known integrals and useful substitutions to make the process faster.