Question

The integral \( \int e^x \sec x (1 + \tan x) \, dx \) is equal to:

• A. \( e^x \sec x + C \)
• B. \( e^x \tan x + C \)
• C. \( e^x (\sec x + \tan x) + C \)
• D. \( e^x \sec x \tan x + C \)
• E. \( e^x \sec x + \tan x + C \)

Hint:

When integrating expressions like \( e^x \sec x \), recognize derivatives within the integrand. In this case, \( 1 + \tan x \) is the derivative of \( \sec x \).

Answer 1

The Correct Option is A

Step 1: Notice that \( 1 + \tan x \) is the derivative of \( \sec x \). So we can rewrite the integral as: \[ \int e^x \frac{d}{dx} (\sec x) \, dx. \] Step 2: This simplifies to: \[ e^x \sec x + C. \]