For this problem, we are asked to rewrite the integral $\int \sec x \, dx$ using substitution.

Part 1: Substitution

We need to determine the correct substitution, $u(x)$, that transforms the integral into the given form: $\int \sec x \, dx = \int \frac{1}{u} \, du$ This suggests that the substitution should result in the differential $\frac{du}{dx}$ being in a simple form like $\sec x$.

A common substitution that simplifies integrals involving $\sec x$ is: $u = \sec x + \tan x$

Now let's compute the differential of $u$: $du = \frac{d}{dx} (\sec x + \tan x) = \sec x \tan x + \sec^2 x \, dx$

This differential, $du = (\sec x \tan x + \sec^2 x) \, dx$, is the correct substitution for part 1.

Part 2: Evaluate the Indefinite Integral

Now that we have the substitution, we can proceed to evaluate the indefinite integral $\int \sec x \, dx$. The formula for this integral is:

$\int \sec x \, dx = \ln | \sec x + \tan x | + C$

So, the answer to part 2 is: $\int \sec x \, dx = \ln | \sec x + \tan x | + C$

Would you like more details on any part of the solution?

Here are 5 related questions to expand your understanding:

1. What is the process of finding substitution in integrals involving trigonometric functions?
2. How do you verify that a substitution is valid for simplifying an integral?
3. What is the integral of $\sec^2 x$?
4. Can substitution be applied to integrals of products of trigonometric functions like $\sec x \cdot \tan x$?
5. How do you find the inverse of trigonometric functions to solve integrals?

Tip: When facing difficult integrals involving trigonometric functions, consider common identities or substitutions (like $u = \sec x + \tan x$) that can simplify the process.