Question

Evaluate: \[ \int \sec x \, (\sec x + \tan x) \, dx \]

Hint:

Remember: $\dfrac{d}{dx}(\tan x) = \sec^2 x$ and $\dfrac{d}{dx}(\sec x) = \sec x \tan x$. These help in quick integration.

Answer 1

Step 1: Expand the integrand.
\[ \int \sec x (\sec x + \tan x) dx = \int \sec^2 x \, dx + \int \sec x \tan x \, dx \]

Step 2: Recall standard integrals.
\[ \int \sec^2 x \, dx = \tan x + C \] \[ \int \sec x \tan x \, dx = \sec x + C \]

Step 3: Combine results.
\[ \int \sec x (\sec x + \tan x) dx = \tan x + \sec x + C \]

Final Answer: \[ \boxed{\tan x + \sec x + C} \]