Question

Evaluate the integral: \[ \int \sqrt{1 + \cos 2x} \, dx = ? \]

• A. \( \sqrt{2} \cos x + c \)
• B. \( \sqrt{2} \sin x + c \)
• C. \( \frac{2}{x^2} + c \)
• D. \( \sqrt{2} \sin \frac{x}{2} + c \)

Hint:

Use trigonometric identities to simplify integrals involving expressions like \( 1 + \cos 2x \). The identity \( 1 + \cos 2x = 2 \cos^2 x \) is especially useful.

Answer 1

The Correct Option is B

Use the identity: \[ 1 + \cos 2x = 2 \cos^2 x \] So: \[ \int \sqrt{1 + \cos 2x} \, dx = \int \sqrt{2 \cos^2 x} \, dx = \int \sqrt{2} |\cos x| \, dx \] Assuming \( \cos x \geq 0 \) in the interval of integration, we get: \[ = \sqrt{2} \int \cos x \, dx = \sqrt{2} \sin x + c \]