Question

Evaluate the integral: \[ \int \frac{e^x \left( 2 + \sin(2x) \right)}{1 + \cos(2x)} \, dx \]

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

Hint:

Remember to apply trigonometric identities to simplify the expression before integrating.

Answer 1

The Correct Option is B

We are given the integral: \[ \int \frac{e^x \left( 2 + \sin(2x) \right)}{1 + \cos(2x)} \, dx \] First, simplify the trigonometric expression using trigonometric identities: \[ 1 + \cos(2x) = 2\cos^2(x) \] Thus, the integral becomes: \[ \int \frac{e^x (2 + \sin(2x))}{2 \cos^2(x)} \, dx \] Now, observe that the integrand simplifies further, and using standard integration techniques, the answer is: \[ e^x \tan x + C \]